Theorem onsucuni3 33215

Description: If an ordinal number has a predecessor, then it is successor of that predecessor. (Contributed by ML, 17-Oct-2020.)

Assertion

Ref	Expression
onsucuni3	|- ( ( B e. On /\ B =/= (/) /\ -. Lim B ) -> B = suc U. B )

Proof of Theorem onsucuni3

Step	Hyp	Ref	Expression
1		eloni 5733	. . . . 5 |- ( B e. On -> Ord B )
2	1	3ad2ant1 1082	. . . 4 |- ( ( B e. On /\ B =/= (/) /\ -. Lim B ) -> Ord B )
3		orduniorsuc 7030	. . . 4 |- ( Ord B -> ( B = U. B \/ B = suc U. B ) )
4	2, 3	syl 17	. . 3 |- ( ( B e. On /\ B =/= (/) /\ -. Lim B ) -> ( B = U. B \/ B = suc U. B ) )
5	4	orcomd 403	. 2 |- ( ( B e. On /\ B =/= (/) /\ -. Lim B ) -> ( B = suc U. B \/ B = U. B ) )
6		simp2 1062	. . 3 |- ( ( B e. On /\ B =/= (/) /\ -. Lim B ) -> B =/= (/) )
7		df-lim 5728	. . . . . . . 8 |- ( Lim B <-> ( Ord B /\ B =/= (/) /\ B = U. B ) )
8	7	biimpri 218	. . . . . . 7 |- ( ( Ord B /\ B =/= (/) /\ B = U. B ) -> Lim B )
9	8	3expb 1266	. . . . . 6 |- ( ( Ord B /\ ( B =/= (/) /\ B = U. B ) ) -> Lim B )
10	9	con3i 150	. . . . 5 |- ( -. Lim B -> -. ( Ord B /\ ( B =/= (/) /\ B = U. B ) ) )
11	10	3ad2ant3 1084	. . . 4 |- ( ( B e. On /\ B =/= (/) /\ -. Lim B ) -> -. ( Ord B /\ ( B =/= (/) /\ B = U. B ) ) )
12	2, 11	mpnanrd 33178	. . 3 |- ( ( B e. On /\ B =/= (/) /\ -. Lim B ) -> -. ( B =/= (/) /\ B = U. B ) )
13	6, 12	mpnanrd 33178	. 2 |- ( ( B e. On /\ B =/= (/) /\ -. Lim B ) -> -. B = U. B )
14		orcom 402	. . 3 |- ( ( B = suc U. B \/ B = U. B ) <-> ( B = U. B \/ B = suc U. B ) )
15		df-or 385	. . 3 |- ( ( B = U. B \/ B = suc U. B ) <-> ( -. B = U. B -> B = suc U. B ) )
16	14, 15	sylbb 209	. 2 |- ( ( B = suc U. B \/ B = U. B ) -> ( -. B = U. B -> B = suc U. B ) )
17	5, 13, 16	sylc 65	1 |- ( ( B e. On /\ B =/= (/) /\ -. Lim B ) -> B = suc U. B )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   (/)c0 3915   U.cuni 4436   Ord word 5722   Oncon0 5723   Lim wlim 5724   suc csuc 5725

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729

This theorem is referenced by:  1oequni2o  33216  rdgsucuni  33217  finxpreclem4  33231