Theorem tfinds 3161

Description: Principle of Transfinite Induction (inference schema) with implicit substitutions. The first four hypotheses establish the substitutions we need. The last three are the basis, the induction hypothesis for successors, and the induction hypothesis for limit ordinals. Theorem Schema 4 of [Suppes] p. 197.

Hypotheses

Ref	Expression
tfinds.1	|- (x = (/) -> (ph <-> ps))
tfinds.2	|- (x = y -> (ph <-> ch))
tfinds.3	|- (x = suc y -> (ph <-> th))
tfinds.4	|- (x = A -> (ph <-> ta))
tfinds.5	|- ps
tfinds.6	|- (y e. On -> (ch -> th))
tfinds.7	|- (Lim x -> (A.y e. x ch -> ph))

Assertion

Ref	Expression
tfinds	|- (A e. On -> ta)

Distinct variable groups:   x,y   x,A   ch,x   ta,x   ph,y

Proof of Theorem tfinds

Step	Hyp	Ref	Expression
1		tfinds.2	. 2 |- (x = y -> (ph <-> ch))
2		tfinds.4	. 2 |- (x = A -> (ph <-> ta))
3		eloni 2958	. . . . 5 |- (x e. On -> Ord x)
4		df-lim 2953	. . . . . . . . . . . . . . . 16 |- (Lim x <-> (Ord x /\ x =/= (/) /\ x = U.x))
5	4	biimpr 152	. . . . . . . . . . . . . . 15 |- ((Ord x /\ x =/= (/) /\ x = U.x) -> Lim x)
6	5	3com23 839	. . . . . . . . . . . . . 14 |- ((Ord x /\ x = U.x /\ x =/= (/)) -> Lim x)
7	6	3expia 835	. . . . . . . . . . . . 13 |- ((Ord x /\ x = U.x) -> (x =/= (/) -> Lim x))
8	7	necon1bd 1632	. . . . . . . . . . . 12 |- ((Ord x /\ x = U.x) -> (-. Lim x -> x = (/)))
9	8	ex 373	. . . . . . . . . . 11 |- (Ord x -> (x = U.x -> (-. Lim x -> x = (/))))
10	9	com23 32	. . . . . . . . . 10 |- (Ord x -> (-. Lim x -> (x = U.x -> x = (/))))
11		orduninsuc 3114	. . . . . . . . . . 11 |- (Ord x -> (x = U.x <-> -. E.y e. On x = suc y))
12	11	biimprd 154	. . . . . . . . . 10 |- (Ord x -> (-. E.y e. On x = suc y -> x = U.x))
13	10, 12	syl5d 55	. . . . . . . . 9 |- (Ord x -> (-. Lim x -> (-. E.y e. On x = suc y -> x = (/))))
14	13	imp 350	. . . . . . . 8 |- ((Ord x /\ -. Lim x) -> (-. E.y e. On x = suc y -> x = (/)))
15	14	con1d 93	. . . . . . 7 |- ((Ord x /\ -. Lim x) -> (-. x = (/) -> E.y e. On x = suc y))
16	15	orrd 233	. . . . . 6 |- ((Ord x /\ -. Lim x) -> (x = (/) \/ E.y e. On x = suc y))
17	16	ex 373	. . . . 5 |- (Ord x -> (-. Lim x -> (x = (/) \/ E.y e. On x = suc y)))
18	3, 17	syl 10	. . . 4 |- (x e. On -> (-. Lim x -> (x = (/) \/ E.y e. On x = suc y)))
19		tfinds.5	. . . . . . 7 |- ps
20		tfinds.1	. . . . . . 7 |- (x = (/) -> (ph <-> ps))
21	19, 20	mpbiri 194	. . . . . 6 |- (x = (/) -> ph)
22	21	a1d 12	. . . . 5 |- (x = (/) -> (A.y e. x ch -> ph))
23		hbra1 1687	. . . . . . 7 |- (A.y e. x ch -> A.yA.y e. x ch)
24		ax-17 971	. . . . . . 7 |- (ph -> A.yph)
25	23, 24	hbim 1007	. . . . . 6 |- ((A.y e. x ch -> ph) -> A.y(A.y e. x ch -> ph))
26		raleq1 1786	. . . . . . . . . . 11 |- (x = suc y -> (A.z e. x [z / x]ph <-> A.z e. suc y[z / x]ph))
27		sbequ 1229	. . . . . . . . . . . . 13 |- (y = z -> ([y / x]ph <-> [z / x]ph))
28		ax-17 971	. . . . . . . . . . . . . 14 |- (ch -> A.xch)
29	28, 1	sbie 1196	. . . . . . . . . . . . 13 |- ([y / x]ph <-> ch)
30	27, 29	syl5bbr 534	. . . . . . . . . . . 12 |- (y = z -> (ch <-> [z / x]ph))
31	30	cbvralv 1800	. . . . . . . . . . 11 |- (A.y e. x ch <-> A.z e. x [z / x]ph)
32		ax-17 971	. . . . . . . . . . . 12 |- (ph -> A.zph)
33		hbs1 1332	. . . . . . . . . . . 12 |- ([z / x]ph -> A.x[z / x]ph)
34		sbequ12 1181	. . . . . . . . . . . 12 |- (x = z -> (ph <-> [z / x]ph))
35	32, 33, 34	cbvral 1798	. . . . . . . . . . 11 |- (A.x e. suc yph <-> A.z e. suc y[z / x]ph)
36	26, 31, 35	3bitr4g 555	. . . . . . . . . 10 |- (x = suc y -> (A.y e. x ch <-> A.x e. suc yph))
37	36	biimpd 153	. . . . . . . . 9 |- (x = suc y -> (A.y e. x ch -> A.x e. suc yph))
38		tfinds.6	. . . . . . . . . 10 |- (y e. On -> (ch -> th))
39		visset 1813	. . . . . . . . . . . 12 |- y e. V
40	39	sucid 3051	. . . . . . . . . . 11 |- y e. suc y
41	1	rcla4v 1873	. . . . . . . . . . 11 |- (y e. suc y -> (A.x e. suc yph -> ch))
42	40, 41	ax-mp 7	. . . . . . . . . 10 |- (A.x e. suc yph -> ch)
43	38, 42	syl5 21	. . . . . . . . 9 |- (y e. On -> (A.x e. suc yph -> th))
44	37, 43	sylan9r 469	. . . . . . . 8 |- ((y e. On /\ x = suc y) -> (A.y e. x ch -> th))
45		tfinds.3	. . . . . . . . 9 |- (x = suc y -> (ph <-> th))
46	45	adantl 388	. . . . . . . 8 |- ((y e. On /\ x = suc y) -> (ph <-> th))
47	44, 46	sylibrd 204	. . . . . . 7 |- ((y e. On /\ x = suc y) -> (A.y e. x ch -> ph))
48	47	ex 373	. . . . . 6 |- (y e. On -> (x = suc y -> (A.y e. x ch -> ph)))
49	25, 48	r19.23ai 1742	. . . . 5 |- (E.y e. On x = suc y -> (A.y e. x ch -> ph))
50	22, 49	jaoi 341	. . . 4 |- ((x = (/) \/ E.y e. On x = suc y) -> (A.y e. x ch -> ph))
51	18, 50	syl6 22	. . 3 |- (x e. On -> (-. Lim x -> (A.y e. x ch -> ph)))
52		tfinds.7	. . 3 |- (Lim x -> (A.y e. x ch -> ph))
53	51, 52	pm2.61d2 129	. 2 |- (x e. On -> (A.y e. x ch -> ph))
54	1, 2, 53	tfis3 3130	1 |- (A e. On -> ta)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958  [wsbc 1170   =/= wne 1585  A.wral 1645  E.wrex 1646  (/)c0 2280  U.cuni 2503  Ord word 2947  Oncon0 2948  Lim wlim 2949  suc csuc 2950

This theorem is referenced by:  tfindsg 3162  tfindes 3164  tfinds3 3166  oa0r 4173  om0r 4174  om1r 4177  oe1m 4179  r1tr 4654  alephon 4865  alephcard 4867  alephordi 4874

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-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

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-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954

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