Theorem sucunielr 4236

Description: Successor and union. The converse (where B is an ordinal) implies excluded middle, as seen at ordsucunielexmid 4256. (Contributed by Jim Kingdon, 2-Aug-2019.)

Assertion

Ref	Expression
sucunielr	|- ( suc A e. B -> A e. U. B )

Proof of Theorem sucunielr

Step	Hyp	Ref	Expression
1		elex 2566	. . . 4 |- ( suc A e. B -> suc A e. _V )
2		sucexb 4223	. . . 4 |- ( A e. _V <-> suc A e. _V )
3	1, 2	sylibr 137	. . 3 |- ( suc A e. B -> A e. _V )
4		sucidg 4153	. . 3 |- ( A e. _V -> A e. suc A )
5	3, 4	syl 14	. 2 |- ( suc A e. B -> A e. suc A )
6		elunii 3585	. 2 |- ( ( A e. suc A /\ suc A e. B ) -> A e. U. B )
7	5, 6	mpancom 399	1 |- ( suc A e. B -> A e. U. B )

Syntax hints:   -> wi 4   e. wcel 1393   _Vcvv 2557   U.cuni 3580   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-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  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-suc 4108

This theorem is referenced by: (None)