Theorem onsucuni2 4612

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 2268	. . . . . 6 |- ( A = suc B -> ( A e. On <-> suc B e. On ) )
2	1	biimpac 298	. . . . 5 |- ( ( A e. On /\ A = suc B ) -> suc B e. On )
3		onsucb 4551	. . . . . . 7 |- ( B e. On <-> suc B e. On )
4		eloni 4422	. . . . . . . . . 10 |- ( B e. On -> Ord B )
5		ordtr 4425	. . . . . . . . . 10 |- ( Ord B -> Tr B )
6	4, 5	syl 14	. . . . . . . . 9 |- ( B e. On -> Tr B )
7		unisucg 4461	. . . . . . . . 9 |- ( B e. On -> ( Tr B <-> U. suc B = B ) )
8	6, 7	mpbid 147	. . . . . . . 8 |- ( B e. On -> U. suc B = B )
9		suceq 4449	. . . . . . . 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 135	. . . . . 6 |- ( suc B e. On -> suc U. suc B = suc B )
12		eloni 4422	. . . . . . . 8 |- ( suc B e. On -> Ord suc B )
13		ordtr 4425	. . . . . . . 8 |- ( Ord suc B -> Tr suc B )
14	12, 13	syl 14	. . . . . . 7 |- ( suc B e. On -> Tr suc B )
15		unisucg 4461	. . . . . . 7 |- ( suc B e. On -> ( Tr suc B <-> U. suc suc B = suc B ) )
16	14, 15	mpbid 147	. . . . . 6 |- ( suc B e. On -> U. suc suc B = suc B )
17	11, 16	eqtr4d 2241	. . . . 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 3859	. . . . . 6 |- ( A = suc B -> U. A = U. suc B )
20		suceq 4449	. . . . . 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 4449	. . . . . 6 |- ( A = suc B -> suc A = suc suc B )
23	22	unieqd 3861	. . . . 5 |- ( A = suc B -> U. suc A = U. suc suc B )
24	21, 23	eqeq12d 2220	. . . 4 |- ( A = suc B -> ( suc U. A = U. suc A <-> suc U. suc B = U. suc suc B ) )
25	18, 24	imbitrrid 156	. . 3 |- ( A = suc B -> ( ( A e. On /\ A = suc B ) -> suc U. A = U. suc A ) )
26	25	anabsi7 581	. 2 |- ( ( A e. On /\ A = suc B ) -> suc U. A = U. suc A )
27		eloni 4422	. . . . 5 |- ( A e. On -> Ord A )
28		ordtr 4425	. . . . 5 |- ( Ord A -> Tr A )
29	27, 28	syl 14	. . . 4 |- ( A e. On -> Tr A )
30		unisucg 4461	. . . 4 |- ( A e. On -> ( Tr A <-> U. suc A = A ) )
31	29, 30	mpbid 147	. . 3 |- ( A e. On -> U. suc A = A )
32	31	adantr 276	. 2 |- ( ( A e. On /\ A = suc B ) -> U. suc A = A )
33	26, 32	eqtrd 2238	1 |- ( ( A e. On /\ A = suc B ) -> suc U. A = A )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2176   U.cuni 3850   Tr wtr 4142   Ord word 4409   Oncon0 4410   suc csuc 4412

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-uni 3851  df-tr 4143  df-iord 4413  df-on 4415  df-suc 4418

This theorem is referenced by:  nnsucpred  4665  nnpredcl  4671