Theorem onsucuni2 4523

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 2220	. . . . . 6 |- ( A = suc B -> ( A e. On <-> suc B e. On ) )
2	1	biimpac 296	. . . . 5 |- ( ( A e. On /\ A = suc B ) -> suc B e. On )
3		sucelon 4462	. . . . . . 7 |- ( B e. On <-> suc B e. On )
4		eloni 4335	. . . . . . . . . 10 |- ( B e. On -> Ord B )
5		ordtr 4338	. . . . . . . . . 10 |- ( Ord B -> Tr B )
6	4, 5	syl 14	. . . . . . . . 9 |- ( B e. On -> Tr B )
7		unisucg 4374	. . . . . . . . 9 |- ( B e. On -> ( Tr B <-> U. suc B = B ) )
8	6, 7	mpbid 146	. . . . . . . 8 |- ( B e. On -> U. suc B = B )
9		suceq 4362	. . . . . . . 8 |- ( U. suc B = B -> suc U. suc B = suc B )
10	8, 9	syl 14	. . . . . . 7 |- ( B e. On -> suc U. suc B = suc B )
11	3, 10	sylbir 134	. . . . . 6 |- ( suc B e. On -> suc U. suc B = suc B )
12		eloni 4335	. . . . . . . 8 |- ( suc B e. On -> Ord suc B )
13		ordtr 4338	. . . . . . . 8 |- ( Ord suc B -> Tr suc B )
14	12, 13	syl 14	. . . . . . 7 |- ( suc B e. On -> Tr suc B )
15		unisucg 4374	. . . . . . 7 |- ( suc B e. On -> ( Tr suc B <-> U. suc suc B = suc B ) )
16	14, 15	mpbid 146	. . . . . 6 |- ( suc B e. On -> U. suc suc B = suc B )
17	11, 16	eqtr4d 2193	. . . . 5 |- ( suc B e. On -> suc U. suc B = U. suc suc B )
18	2, 17	syl 14	. . . 4 |- ( ( A e. On /\ A = suc B ) -> suc U. suc B = U. suc suc B )
19		unieq 3781	. . . . . 6 |- ( A = suc B -> U. A = U. suc B )
20		suceq 4362	. . . . . 6 |- ( U. A = U. suc B -> suc U. A = suc U. suc B )
21	19, 20	syl 14	. . . . 5 |- ( A = suc B -> suc U. A = suc U. suc B )
22		suceq 4362	. . . . . 6 |- ( A = suc B -> suc A = suc suc B )
23	22	unieqd 3783	. . . . 5 |- ( A = suc B -> U. suc A = U. suc suc B )
24	21, 23	eqeq12d 2172	. . . 4 |- ( A = suc B -> ( suc U. A = U. suc A <-> suc U. suc B = U. suc suc B ) )
25	18, 24	syl5ibr 155	. . 3 |- ( A = suc B -> ( ( A e. On /\ A = suc B ) -> suc U. A = U. suc A ) )
26	25	anabsi7 571	. 2 |- ( ( A e. On /\ A = suc B ) -> suc U. A = U. suc A )
27		eloni 4335	. . . . 5 |- ( A e. On -> Ord A )
28		ordtr 4338	. . . . 5 |- ( Ord A -> Tr A )
29	27, 28	syl 14	. . . 4 |- ( A e. On -> Tr A )
30		unisucg 4374	. . . 4 |- ( A e. On -> ( Tr A <-> U. suc A = A ) )
31	29, 30	mpbid 146	. . 3 |- ( A e. On -> U. suc A = A )
32	31	adantr 274	. 2 |- ( ( A e. On /\ A = suc B ) -> U. suc A = A )
33	26, 32	eqtrd 2190	1 |- ( ( A e. On /\ A = suc B ) -> suc U. A = A )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1335   e. wcel 2128   U.cuni 3772   Tr wtr 4062   Ord word 4322   Oncon0 4323   suc csuc 4325

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-uni 3773  df-tr 4063  df-iord 4326  df-on 4328  df-suc 4331

This theorem is referenced by:  nnsucpred  4576  nnpredcl  4582