Theorem onsucuni2 4380

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 2150	. . . . . 6 |- ( A = suc B -> ( A e. On <-> suc B e. On ) )
2	1	biimpac 292	. . . . 5 |- ( ( A e. On /\ A = suc B ) -> suc B e. On )
3		sucelon 4320	. . . . . . 7 |- ( B e. On <-> suc B e. On )
4		eloni 4202	. . . . . . . . . 10 |- ( B e. On -> Ord B )
5		ordtr 4205	. . . . . . . . . 10 |- ( Ord B -> Tr B )
6	4, 5	syl 14	. . . . . . . . 9 |- ( B e. On -> Tr B )
7		unisucg 4241	. . . . . . . . 9 |- ( B e. On -> ( Tr B <-> U. suc B = B ) )
8	6, 7	mpbid 145	. . . . . . . 8 |- ( B e. On -> U. suc B = B )
9		suceq 4229	. . . . . . . 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 133	. . . . . 6 |- ( suc B e. On -> suc U. suc B = suc B )
12		eloni 4202	. . . . . . . 8 |- ( suc B e. On -> Ord suc B )
13		ordtr 4205	. . . . . . . 8 |- ( Ord suc B -> Tr suc B )
14	12, 13	syl 14	. . . . . . 7 |- ( suc B e. On -> Tr suc B )
15		unisucg 4241	. . . . . . 7 |- ( suc B e. On -> ( Tr suc B <-> U. suc suc B = suc B ) )
16	14, 15	mpbid 145	. . . . . 6 |- ( suc B e. On -> U. suc suc B = suc B )
17	11, 16	eqtr4d 2123	. . . . 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 3662	. . . . . 6 |- ( A = suc B -> U. A = U. suc B )
20		suceq 4229	. . . . . 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 4229	. . . . . 6 |- ( A = suc B -> suc A = suc suc B )
23	22	unieqd 3664	. . . . 5 |- ( A = suc B -> U. suc A = U. suc suc B )
24	21, 23	eqeq12d 2102	. . . 4 |- ( A = suc B -> ( suc U. A = U. suc A <-> suc U. suc B = U. suc suc B ) )
25	18, 24	syl5ibr 154	. . 3 |- ( A = suc B -> ( ( A e. On /\ A = suc B ) -> suc U. A = U. suc A ) )
26	25	anabsi7 548	. 2 |- ( ( A e. On /\ A = suc B ) -> suc U. A = U. suc A )
27		eloni 4202	. . . . 5 |- ( A e. On -> Ord A )
28		ordtr 4205	. . . . 5 |- ( Ord A -> Tr A )
29	27, 28	syl 14	. . . 4 |- ( A e. On -> Tr A )
30		unisucg 4241	. . . 4 |- ( A e. On -> ( Tr A <-> U. suc A = A ) )
31	29, 30	mpbid 145	. . 3 |- ( A e. On -> U. suc A = A )
32	31	adantr 270	. 2 |- ( ( A e. On /\ A = suc B ) -> U. suc A = A )
33	26, 32	eqtrd 2120	1 |- ( ( A e. On /\ A = suc B ) -> suc U. A = A )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1289   e. wcel 1438   U.cuni 3653   Tr wtr 3936   Ord word 4189   Oncon0 4190   suc csuc 4192

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-uni 3654  df-tr 3937  df-iord 4193  df-on 4195  df-suc 4198

This theorem is referenced by:  nnpredcl  11845  nnsucpred  11846