Theorem ordunisuc2 3110

Description: An ordinal equal to its union contains the successor of each of its members.

Assertion

Ref	Expression
ordunisuc2	|- (Ord A -> (A = U.A <-> A.x e. A suc x e. A))

Distinct variable group:   x,A

Proof of Theorem ordunisuc2

Step	Hyp	Ref	Expression
1		orduninsuc 3109	. 2 |- (Ord A -> (A = U.A <-> -. E.x e. On A = suc x))
2		ordtri3 2978	. . . . . . . . 9 |- ((Ord A /\ Ord suc x) -> (A = suc x <-> -. (A e. suc x \/ suc x e. A)))
3		suceloni 3057	. . . . . . . . . 10 |- (x e. On -> suc x e. On)
4		eloni 2953	. . . . . . . . . 10 |- (suc x e. On -> Ord suc x)
5	3, 4	syl 10	. . . . . . . . 9 |- (x e. On -> Ord suc x)
6	2, 5	sylan2 451	. . . . . . . 8 |- ((Ord A /\ x e. On) -> (A = suc x <-> -. (A e. suc x \/ suc x e. A)))
7	6	con2bid 525	. . . . . . 7 |- ((Ord A /\ x e. On) -> ((A e. suc x \/ suc x e. A) <-> -. A = suc x))
8		onnbtwn 3059	. . . . . . . . . . . . 13 |- (x e. On -> -. (x e. A /\ A e. suc x))
9		imnan 242	. . . . . . . . . . . . 13 |- ((x e. A -> -. A e. suc x) <-> -. (x e. A /\ A e. suc x))
10	8, 9	sylibr 200	. . . . . . . . . . . 12 |- (x e. On -> (x e. A -> -. A e. suc x))
11	10	con2d 91	. . . . . . . . . . 11 |- (x e. On -> (A e. suc x -> -. x e. A))
12		pm2.21 76	. . . . . . . . . . 11 |- (-. x e. A -> (x e. A -> suc x e. A))
13	11, 12	syl6 22	. . . . . . . . . 10 |- (x e. On -> (A e. suc x -> (x e. A -> suc x e. A)))
14	13	adantl 388	. . . . . . . . 9 |- ((Ord A /\ x e. On) -> (A e. suc x -> (x e. A -> suc x e. A)))
15		ax-1 4	. . . . . . . . . 10 |- (suc x e. A -> (x e. A -> suc x e. A))
16	15	a1i 8	. . . . . . . . 9 |- ((Ord A /\ x e. On) -> (suc x e. A -> (x e. A -> suc x e. A)))
17	14, 16	jaod 424	. . . . . . . 8 |- ((Ord A /\ x e. On) -> ((A e. suc x \/ suc x e. A) -> (x e. A -> suc x e. A)))
18		ordtri2or 3072	. . . . . . . . . . . . . 14 |- ((Ord x /\ Ord A) -> (x e. A \/ A (_ x))
19		eloni 2953	. . . . . . . . . . . . . 14 |- (x e. On -> Ord x)
20	18, 19	sylan 448	. . . . . . . . . . . . 13 |- ((x e. On /\ Ord A) -> (x e. A \/ A (_ x))
21	20	ancoms 436	. . . . . . . . . . . 12 |- ((Ord A /\ x e. On) -> (x e. A \/ A (_ x))
22		orcom 246	. . . . . . . . . . . 12 |- ((x e. A \/ A (_ x) <-> (A (_ x \/ x e. A))
23	21, 22	sylib 198	. . . . . . . . . . 11 |- ((Ord A /\ x e. On) -> (A (_ x \/ x e. A))
24	23	adantr 389	. . . . . . . . . 10 |- (((Ord A /\ x e. On) /\ (x e. A -> suc x e. A)) -> (A (_ x \/ x e. A))
25		ordsssuc2 3054	. . . . . . . . . . . . 13 |- ((Ord A /\ x e. On) -> (A (_ x <-> A e. suc x))
26	25	biimpd 153	. . . . . . . . . . . 12 |- ((Ord A /\ x e. On) -> (A (_ x -> A e. suc x))
27	26	adantr 389	. . . . . . . . . . 11 |- (((Ord A /\ x e. On) /\ (x e. A -> suc x e. A)) -> (A (_ x -> A e. suc x))
28		pm3.27 323	. . . . . . . . . . 11 |- (((Ord A /\ x e. On) /\ (x e. A -> suc x e. A)) -> (x e. A -> suc x e. A))
29	27, 28	orim12d 564	. . . . . . . . . 10 |- (((Ord A /\ x e. On) /\ (x e. A -> suc x e. A)) -> ((A (_ x \/ x e. A) -> (A e. suc x \/ suc x e. A)))
30	24, 29	mpd 26	. . . . . . . . 9 |- (((Ord A /\ x e. On) /\ (x e. A -> suc x e. A)) -> (A e. suc x \/ suc x e. A))
31	30	ex 373	. . . . . . . 8 |- ((Ord A /\ x e. On) -> ((x e. A -> suc x e. A) -> (A e. suc x \/ suc x e. A)))
32	17, 31	impbid 515	. . . . . . 7 |- ((Ord A /\ x e. On) -> ((A e. suc x \/ suc x e. A) <-> (x e. A -> suc x e. A)))
33	7, 32	bitr3d 529	. . . . . 6 |- ((Ord A /\ x e. On) -> (-. A = suc x <-> (x e. A -> suc x e. A)))
34	33	pm5.74da 585	. . . . 5 |- (Ord A -> ((x e. On -> -. A = suc x) <-> (x e. On -> (x e. A -> suc x e. A))))
35		pm3.27 323	. . . . . . . 8 |- ((x e. On /\ x e. A) -> x e. A)
36		ordelon 2966	. . . . . . . . . 10 |- ((Ord A /\ x e. A) -> x e. On)
37	36	ex 373	. . . . . . . . 9 |- (Ord A -> (x e. A -> x e. On))
38	37	ancrd 299	. . . . . . . 8 |- (Ord A -> (x e. A -> (x e. On /\ x e. A)))
39	35, 38	impbid2 517	. . . . . . 7 |- (Ord A -> ((x e. On /\ x e. A) <-> x e. A))
40	39	imbi1d 612	. . . . . 6 |- (Ord A -> (((x e. On /\ x e. A) -> suc x e. A) <-> (x e. A -> suc x e. A)))
41		impexp 347	. . . . . 6 |- (((x e. On /\ x e. A) -> suc x e. A) <-> (x e. On -> (x e. A -> suc x e. A)))
42	40, 41	syl5bbr 533	. . . . 5 |- (Ord A -> ((x e. On -> (x e. A -> suc x e. A)) <-> (x e. A -> suc x e. A)))
43	34, 42	bitrd 527	. . . 4 |- (Ord A -> ((x e. On -> -. A = suc x) <-> (x e. A -> suc x e. A)))
44	43	ralbidv2 1662	. . 3 |- (Ord A -> (A.x e. On -. A = suc x <-> A.x e. A suc x e. A))
45		ralnex 1650	. . 3 |- (A.x e. On -. A = suc x <-> -. E.x e. On A = suc x)
46	44, 45	syl5bbr 533	. 2 |- (Ord A -> (-. E.x e. On A = suc x <-> A.x e. A suc x e. A))
47	1, 46	bitrd 527	1 |- (Ord A -> (A = U.A <-> A.x e. A suc x e. A))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   = wceq 954   e. wcel 956  A.wral 1642  E.wrex 1643   (_ wss 2043  U.cuni 2498  Ord word 2942  Oncon0 2943  suc csuc 2945

This theorem is referenced by:  dflim4 3114

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-suc 2949

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