Theorem sucunielr 4487

Description: Successor and union. The converse (where B is an ordinal) implies excluded middle, as seen at ordsucunielexmid 4508. (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 2737	. . . 4 |- ( suc A e. B -> suc A e. _V )
2		sucexb 4474	. . . 4 |- ( A e. _V <-> suc A e. _V )
3	1, 2	sylibr 133	. . 3 |- ( suc A e. B -> A e. _V )
4		sucidg 4394	. . 3 |- ( A e. _V -> A e. suc A )
5	3, 4	syl 14	. 2 |- ( suc A e. B -> A e. suc A )
6		elunii 3794	. 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 2136   _Vcvv 2726   U.cuni 3789   suc csuc 4343

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-uni 3790  df-suc 4349

This theorem is referenced by:  nnsucuniel  6463