Theorem ordunisuc 7032

Description: An ordinal class is equal to the union of its successor. (Contributed by NM, 10-Dec-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
ordunisuc	|- ( Ord A -> U. suc A = A )

Proof of Theorem ordunisuc

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ordeleqon 6988	. 2 |- ( Ord A <-> ( A e. On \/ A = On ) )
2		suceq 5790	. . . . . 6 |- ( x = A -> suc x = suc A )
3	2	unieqd 4446	. . . . 5 |- ( x = A -> U. suc x = U. suc A )
4		id 22	. . . . 5 |- ( x = A -> x = A )
5	3, 4	eqeq12d 2637	. . . 4 |- ( x = A -> ( U. suc x = x <-> U. suc A = A ) )
6		eloni 5733	. . . . . 6 |- ( x e. On -> Ord x )
7		ordtr 5737	. . . . . 6 |- ( Ord x -> Tr x )
8	6, 7	syl 17	. . . . 5 |- ( x e. On -> Tr x )
9		vex 3203	. . . . . 6 |- x e. _V
10	9	unisuc 5801	. . . . 5 |- ( Tr x <-> U. suc x = x )
11	8, 10	sylib 208	. . . 4 |- ( x e. On -> U. suc x = x )
12	5, 11	vtoclga 3272	. . 3 |- ( A e. On -> U. suc A = A )
13		sucon 7008	. . . . . 6 |- suc On = On
14	13	unieqi 4445	. . . . 5 |- U. suc On = U. On
15		unon 7031	. . . . 5 |- U. On = On
16	14, 15	eqtri 2644	. . . 4 |- U. suc On = On
17		suceq 5790	. . . . 5 |- ( A = On -> suc A = suc On )
18	17	unieqd 4446	. . . 4 |- ( A = On -> U. suc A = U. suc On )
19		id 22	. . . 4 |- ( A = On -> A = On )
20	16, 18, 19	3eqtr4a 2682	. . 3 |- ( A = On -> U. suc A = A )
21	12, 20	jaoi 394	. 2 |- ( ( A e. On \/ A = On ) -> U. suc A = A )
22	1, 21	sylbi 207	1 |- ( Ord A -> U. suc A = A )

Syntax hints:   -> wi 4   \/ wo 383   = wceq 1483   e. wcel 1990   U.cuni 4436   Tr wtr 4752   Ord word 5722   Oncon0 5723   suc csuc 5725

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727  df-suc 5729

This theorem is referenced by:  orduniss2  7033  onsucuni2  7034  nlimsucg  7042  tz7.44-2  7503  ttukeylem7  9337  tsksuc  9584  dfrdg2  31701  ontgsucval  32431  onsuctopon  32433  limsucncmpi  32444  finxpsuclem  33234