Theorem zfregs 4647

Description: The strong form of the Axiom of Regularity, which does not require that A be a set. Axiom 6' of [TakeutiZaring] p. 21. The proof makes use of a special case of Proposition 9.4 of [TakeutiZaring] p. 75.

Assertion

Ref	Expression
zfregs	|- (A =/= (/) -> E.x e. A (x i^i A) = (/))

Distinct variable group:   x,A

Proof of Theorem zfregs

Step	Hyp	Ref	Expression
1		ne0 2288	. 2 |- (A =/= (/) <-> E.z z e. A)
2		snex 2750	. . . . 5 |- {z} e. V
3	2	tz9.1 4646	. . . 4 |- E.y({z} (_ y /\ Tr y /\ A.w(({z} (_ w /\ Tr w) -> y (_ w))
4		trel 2687	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (Tr y -> ((w e. x /\ x e. y) -> w e. y))
5		inass 2223	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ((y i^i A) i^i x) = (y i^i (A i^i x))
6		incom 2208	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- (A i^i x) = (x i^i A)
7	6	ineq2i 2214	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- (y i^i (A i^i x)) = (y i^i (x i^i A))
8	5, 7	eqtr 1495	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ((y i^i A) i^i x) = (y i^i (x i^i A))
9	8	eleq2i 1538	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- (w e. ((y i^i A) i^i x) <-> w e. (y i^i (x i^i A)))
10		elin 2207	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- (w e. (y i^i (x i^i A)) <-> (w e. y /\ w e. (x i^i A)))
11	9, 10	bitr2 174	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ((w e. y /\ w e. (x i^i A)) <-> w e. ((y i^i A) i^i x))
12		ne0i 2286	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (w e. ((y i^i A) i^i x) -> ((y i^i A) i^i x) =/= (/))
13	11, 12	sylbi 199	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ((w e. y /\ w e. (x i^i A)) -> ((y i^i A) i^i x) =/= (/))
14	13	ex 373	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (w e. y -> (w e. (x i^i A) -> ((y i^i A) i^i x) =/= (/)))
15	4, 14	syl6 22	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (Tr y -> ((w e. x /\ x e. y) -> (w e. (x i^i A) -> ((y i^i A) i^i x) =/= (/))))
16	15	exp3a 375	. . . . . . . . . . . . . . . . . . . . . . 23 |- (Tr y -> (w e. x -> (x e. y -> (w e. (x i^i A) -> ((y i^i A) i^i x) =/= (/)))))
17	16	com34 36	. . . . . . . . . . . . . . . . . . . . . 22 |- (Tr y -> (w e. x -> (w e. (x i^i A) -> (x e. y -> ((y i^i A) i^i x) =/= (/)))))
18	17	imp3a 361	. . . . . . . . . . . . . . . . . . . . 21 |- (Tr y -> ((w e. x /\ w e. (x i^i A)) -> (x e. y -> ((y i^i A) i^i x) =/= (/))))
19		inss1 2230	. . . . . . . . . . . . . . . . . . . . . . 23 |- (x i^i A) (_ x
20	19	sseli 2065	. . . . . . . . . . . . . . . . . . . . . 22 |- (w e. (x i^i A) -> w e. x)
21	20	ancri 297	. . . . . . . . . . . . . . . . . . . . 21 |- (w e. (x i^i A) -> (w e. x /\ w e. (x i^i A)))
22	18, 21	syl5 21	. . . . . . . . . . . . . . . . . . . 20 |- (Tr y -> (w e. (x i^i A) -> (x e. y -> ((y i^i A) i^i x) =/= (/))))
23	22	19.23adv 1214	. . . . . . . . . . . . . . . . . . 19 |- (Tr y -> (E.w w e. (x i^i A) -> (x e. y -> ((y i^i A) i^i x) =/= (/))))
24		ne0 2288	. . . . . . . . . . . . . . . . . . 19 |- ((x i^i A) =/= (/) <-> E.w w e. (x i^i A))
25	23, 24	syl5ib 206	. . . . . . . . . . . . . . . . . 18 |- (Tr y -> ((x i^i A) =/= (/) -> (x e. y -> ((y i^i A) i^i x) =/= (/))))
26	25	com23 32	. . . . . . . . . . . . . . . . 17 |- (Tr y -> (x e. y -> ((x i^i A) =/= (/) -> ((y i^i A) i^i x) =/= (/))))
27	26	imp 350	. . . . . . . . . . . . . . . 16 |- ((Tr y /\ x e. y) -> ((x i^i A) =/= (/) -> ((y i^i A) i^i x) =/= (/)))
28	27	necon4d 1628	. . . . . . . . . . . . . . 15 |- ((Tr y /\ x e. y) -> (((y i^i A) i^i x) = (/) -> (x i^i A) = (/)))
29	28	anim2d 561	. . . . . . . . . . . . . 14 |- ((Tr y /\ x e. y) -> ((x e. A /\ ((y i^i A) i^i x) = (/)) -> (x e. A /\ (x i^i A) = (/))))
30	29	ex 373	. . . . . . . . . . . . 13 |- (Tr y -> (x e. y -> ((x e. A /\ ((y i^i A) i^i x) = (/)) -> (x e. A /\ (x i^i A) = (/)))))
31	30	imp3a 361	. . . . . . . . . . . 12 |- (Tr y -> ((x e. y /\ (x e. A /\ ((y i^i A) i^i x) = (/))) -> (x e. A /\ (x i^i A) = (/))))
32		elin 2207	. . . . . . . . . . . . . 14 |- (x e. (y i^i A) <-> (x e. y /\ x e. A))
33	32	anbi1i 481	. . . . . . . . . . . . 13 |- ((x e. (y i^i A) /\ ((y i^i A) i^i x) = (/)) <-> ((x e. y /\ x e. A) /\ ((y i^i A) i^i x) = (/)))
34		anass 439	. . . . . . . . . . . . 13 |- (((x e. y /\ x e. A) /\ ((y i^i A) i^i x) = (/)) <-> (x e. y /\ (x e. A /\ ((y i^i A) i^i x) = (/))))
35	33, 34	bitr 173	. . . . . . . . . . . 12 |- ((x e. (y i^i A) /\ ((y i^i A) i^i x) = (/)) <-> (x e. y /\ (x e. A /\ ((y i^i A) i^i x) = (/))))
36	31, 35	syl5ib 206	. . . . . . . . . . 11 |- (Tr y -> ((x e. (y i^i A) /\ ((y i^i A) i^i x) = (/)) -> (x e. A /\ (x i^i A) = (/))))
37	36	r19.22dv2 1736	. . . . . . . . . 10 |- (Tr y -> (E.x e. (y i^i A)((y i^i A) i^i x) = (/) -> E.x e. A (x i^i A) = (/)))
38		visset 1813	. . . . . . . . . . . 12 |- y e. V
39	38	inex1 2716	. . . . . . . . . . 11 |- (y i^i A) e. V
40	39	zfreg2 4597	. . . . . . . . . 10 |- ((y i^i A) =/= (/) -> E.x e. (y i^i A)((y i^i A) i^i x) = (/))
41	37, 40	syl5 21	. . . . . . . . 9 |- (Tr y -> ((y i^i A) =/= (/) -> E.x e. A (x i^i A) = (/)))
42		snssi 2466	. . . . . . . . . . . 12 |- (z e. A -> {z} (_ A)
43	42	anim2i 335	. . . . . . . . . . 11 |- (({z} (_ y /\ z e. A) -> ({z} (_ y /\ {z} (_ A))
44		ssin 2232	. . . . . . . . . . . 12 |- (({z} (_ y /\ {z} (_ A) <-> {z} (_ (y i^i A))
45		visset 1813	. . . . . . . . . . . . 13 |- z e. V
46	45	snss 2461	. . . . . . . . . . . 12 |- (z e. (y i^i A) <-> {z} (_ (y i^i A))
47	44, 46	bitr4 176	. . . . . . . . . . 11 |- (({z} (_ y /\ {z} (_ A) <-> z e. (y i^i A))
48	43, 47	sylib 198	. . . . . . . . . 10 |- (({z} (_ y /\ z e. A) -> z e. (y i^i A))
49		ne0i 2286	. . . . . . . . . 10 |- (z e. (y i^i A) -> (y i^i A) =/= (/))
50	48, 49	syl 10	. . . . . . . . 9 |- (({z} (_ y /\ z e. A) -> (y i^i A) =/= (/))
51	41, 50	syl5 21	. . . . . . . 8 |- (Tr y -> (({z} (_ y /\ z e. A) -> E.x e. A (x i^i A) = (/)))
52	51	exp3a 375	. . . . . . 7 |- (Tr y -> ({z} (_ y -> (z e. A -> E.x e. A (x i^i A) = (/))))
53	52	impcom 351	. . . . . 6 |- (({z} (_ y /\ Tr y) -> (z e. A -> E.x e. A (x i^i A) = (/)))
54	53	3adant3 799	. . . . 5 |- (({z} (_ y /\ Tr y /\ A.w(({z} (_ w /\ Tr w) -> y (_ w)) -> (z e. A -> E.x e. A (x i^i A) = (/)))
55	54	19.23aiv 1295	. . . 4 |- (E.y({z} (_ y /\ Tr y /\ A.w(({z} (_ w /\ Tr w) -> y (_ w)) -> (z e. A -> E.x e. A (x i^i A) = (/)))
56	3, 55	ax-mp 7	. . 3 |- (z e. A -> E.x e. A (x i^i A) = (/))
57	56	19.23aiv 1295	. 2 |- (E.z z e. A -> E.x e. A (x i^i A) = (/))
58	1, 57	sylbi 199	1 |- (A =/= (/) -> E.x e. A (x i^i A) = (/))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 775  A.wal 954   = wceq 956   e. wcel 958  E.wex 980   =/= wne 1585  E.wrex 1646   i^i cin 2046   (_ wss 2047  (/)c0 2280  {csn 2409  Tr wtr 2680

This theorem is referenced by:  setind 4648

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-reg 4593  ax-inf2 4625

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-sbc 1942  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-fv 3198  df-rdg 3932

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