Theorem sucunielr 4434

Description: Successor and union. The converse (where B is an ordinal) implies excluded middle, as seen at ordsucunielexmid 4454. (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 2700	. . . 4 |- ( suc A e. B -> suc A e. _V )
2		sucexb 4421	. . . 4 |- ( A e. _V <-> suc A e. _V )
3	1, 2	sylibr 133	. . 3 |- ( suc A e. B -> A e. _V )
4		sucidg 4346	. . 3 |- ( A e. _V -> A e. suc A )
5	3, 4	syl 14	. 2 |- ( suc A e. B -> A e. suc A )
6		elunii 3749	. 2 |- ( ( A e. suc A /\ suc A e. B ) -> A e. U. B )
7	5, 6	mpancom 419	1 |- ( suc A e. B -> A e. U. B )

Syntax hints:   -> wi 4   e. wcel 1481   _Vcvv 2689   U.cuni 3744   suc csuc 4295

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-uni 3745  df-suc 4301

This theorem is referenced by:  nnsucuniel  6399