Theorem onsucuni2 4541

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 2229	. . . . . 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 4480	. . . . . . 7 |- ( B e. On <-> suc B e. On )
4		eloni 4353	. . . . . . . . . 10 |- ( B e. On -> Ord B )
5		ordtr 4356	. . . . . . . . . 10 |- ( Ord B -> Tr B )
6	4, 5	syl 14	. . . . . . . . 9 |- ( B e. On -> Tr B )
7		unisucg 4392	. . . . . . . . 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 4380	. . . . . . . 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 4353	. . . . . . . 8 |- ( suc B e. On -> Ord suc B )
13		ordtr 4356	. . . . . . . 8 |- ( Ord suc B -> Tr suc B )
14	12, 13	syl 14	. . . . . . 7 |- ( suc B e. On -> Tr suc B )
15		unisucg 4392	. . . . . . 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 2201	. . . . 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 3798	. . . . . 6 |- ( A = suc B -> U. A = U. suc B )
20		suceq 4380	. . . . . 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 4380	. . . . . 6 |- ( A = suc B -> suc A = suc suc B )
23	22	unieqd 3800	. . . . 5 |- ( A = suc B -> U. suc A = U. suc suc B )
24	21, 23	eqeq12d 2180	. . . 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 4353	. . . . 5 |- ( A e. On -> Ord A )
28		ordtr 4356	. . . . 5 |- ( Ord A -> Tr A )
29	27, 28	syl 14	. . . 4 |- ( A e. On -> Tr A )
30		unisucg 4392	. . . 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 2198	1 |- ( ( A e. On /\ A = suc B ) -> suc U. A = A )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   U.cuni 3789   Tr wtr 4080   Ord word 4340   Oncon0 4341   suc csuc 4343

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-uni 3790  df-tr 4081  df-iord 4344  df-on 4346  df-suc 4349

This theorem is referenced by:  nnsucpred  4594  nnpredcl  4600