Theorem onsucuni2 7034

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

Assertion

Ref	Expression
onsucuni2	|- ( ( A e. On /\ A = suc B ) -> suc U. A = A )

Proof of Theorem onsucuni2

Step	Hyp	Ref	Expression
1		eleq1 2689	. . . . . 6 |- ( A = suc B -> ( A e. On <-> suc B e. On ) )
2	1	biimpac 503	. . . . 5 |- ( ( A e. On /\ A = suc B ) -> suc B e. On )
3		eloni 5733	. . . . 5 |- ( suc B e. On -> Ord suc B )
4		ordsuc 7014	. . . . . . . 8 |- ( Ord B <-> Ord suc B )
5		ordunisuc 7032	. . . . . . . 8 |- ( Ord B -> U. suc B = B )
6	4, 5	sylbir 225	. . . . . . 7 |- ( Ord suc B -> U. suc B = B )
7		suceq 5790	. . . . . . 7 |- ( U. suc B = B -> suc U. suc B = suc B )
8	6, 7	syl 17	. . . . . 6 |- ( Ord suc B -> suc U. suc B = suc B )
9		ordunisuc 7032	. . . . . 6 |- ( Ord suc B -> U. suc suc B = suc B )
10	8, 9	eqtr4d 2659	. . . . 5 |- ( Ord suc B -> suc U. suc B = U. suc suc B )
11	2, 3, 10	3syl 18	. . . 4 |- ( ( A e. On /\ A = suc B ) -> suc U. suc B = U. suc suc B )
12		unieq 4444	. . . . . 6 |- ( A = suc B -> U. A = U. suc B )
13		suceq 5790	. . . . . 6 |- ( U. A = U. suc B -> suc U. A = suc U. suc B )
14	12, 13	syl 17	. . . . 5 |- ( A = suc B -> suc U. A = suc U. suc B )
15		suceq 5790	. . . . . 6 |- ( A = suc B -> suc A = suc suc B )
16	15	unieqd 4446	. . . . 5 |- ( A = suc B -> U. suc A = U. suc suc B )
17	14, 16	eqeq12d 2637	. . . 4 |- ( A = suc B -> ( suc U. A = U. suc A <-> suc U. suc B = U. suc suc B ) )
18	11, 17	syl5ibr 236	. . 3 |- ( A = suc B -> ( ( A e. On /\ A = suc B ) -> suc U. A = U. suc A ) )
19	18	anabsi7 860	. 2 |- ( ( A e. On /\ A = suc B ) -> suc U. A = U. suc A )
20		eloni 5733	. . . 4 |- ( A e. On -> Ord A )
21		ordunisuc 7032	. . . 4 |- ( Ord A -> U. suc A = A )
22	20, 21	syl 17	. . 3 |- ( A e. On -> U. suc A = A )
23	22	adantr 481	. 2 |- ( ( A e. On /\ A = suc B ) -> U. suc A = A )
24	19, 23	eqtrd 2656	1 |- ( ( A e. On /\ A = suc B ) -> suc U. A = A )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   U.cuni 4436   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:  rankxplim3  8744  rankxpsuc  8745