Theorem onsucuni2 4288

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 2100	. . . . . 6 |- ( A = suc B -> ( A e. On <-> suc B e. On ) )
2	1	biimpac 282	. . . . 5 |- ( ( A e. On /\ A = suc B ) -> suc B e. On )
3		sucelon 4229	. . . . . . 7 |- ( B e. On <-> suc B e. On )
4		eloni 4112	. . . . . . . . . 10 |- ( B e. On -> Ord B )
5		ordtr 4115	. . . . . . . . . 10 |- ( Ord B -> Tr B )
6	4, 5	syl 14	. . . . . . . . 9 |- ( B e. On -> Tr B )
7		unisucg 4151	. . . . . . . . 9 |- ( B e. On -> ( Tr B <-> U. suc B = B ) )
8	6, 7	mpbid 135	. . . . . . . 8 |- ( B e. On -> U. suc B = B )
9		suceq 4139	. . . . . . . 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 125	. . . . . 6 |- ( suc B e. On -> suc U. suc B = suc B )
12		eloni 4112	. . . . . . . 8 |- ( suc B e. On -> Ord suc B )
13		ordtr 4115	. . . . . . . 8 |- ( Ord suc B -> Tr suc B )
14	12, 13	syl 14	. . . . . . 7 |- ( suc B e. On -> Tr suc B )
15		unisucg 4151	. . . . . . 7 |- ( suc B e. On -> ( Tr suc B <-> U. suc suc B = suc B ) )
16	14, 15	mpbid 135	. . . . . 6 |- ( suc B e. On -> U. suc suc B = suc B )
17	11, 16	eqtr4d 2075	. . . . 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 3589	. . . . . 6 |- ( A = suc B -> U. A = U. suc B )
20		suceq 4139	. . . . . 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 4139	. . . . . 6 |- ( A = suc B -> suc A = suc suc B )
23	22	unieqd 3591	. . . . 5 |- ( A = suc B -> U. suc A = U. suc suc B )
24	21, 23	eqeq12d 2054	. . . 4 |- ( A = suc B -> ( suc U. A = U. suc A <-> suc U. suc B = U. suc suc B ) )
25	18, 24	syl5ibr 145	. . 3 |- ( A = suc B -> ( ( A e. On /\ A = suc B ) -> suc U. A = U. suc A ) )
26	25	anabsi7 515	. 2 |- ( ( A e. On /\ A = suc B ) -> suc U. A = U. suc A )
27		eloni 4112	. . . . 5 |- ( A e. On -> Ord A )
28		ordtr 4115	. . . . 5 |- ( Ord A -> Tr A )
29	27, 28	syl 14	. . . 4 |- ( A e. On -> Tr A )
30		unisucg 4151	. . . 4 |- ( A e. On -> ( Tr A <-> U. suc A = A ) )
31	29, 30	mpbid 135	. . 3 |- ( A e. On -> U. suc A = A )
32	31	adantr 261	. 2 |- ( ( A e. On /\ A = suc B ) -> U. suc A = A )
33	26, 32	eqtrd 2072	1 |- ( ( A e. On /\ A = suc B ) -> suc U. A = A )

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1243   e. wcel 1393   U.cuni 3580   Tr wtr 3854   Ord word 4099   Oncon0 4100   suc csuc 4102

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-uni 3581  df-tr 3855  df-iord 4103  df-on 4105  df-suc 4108

This theorem is referenced by: (None)