Theorem sucunielr 4494

Description: Successor and union. The converse (where B is an ordinal) implies excluded middle, as seen at ordsucunielexmid 4515. (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 2741	. . . 4 |- ( suc A e. B -> suc A e. _V )
2		sucexb 4481	. . . 4 |- ( A e. _V <-> suc A e. _V )
3	1, 2	sylibr 133	. . 3 |- ( suc A e. B -> A e. _V )
4		sucidg 4401	. . 3 |- ( A e. _V -> A e. suc A )
5	3, 4	syl 14	. 2 |- ( suc A e. B -> A e. suc A )
6		elunii 3801	. 2 |- ( ( A e. suc A /\ suc A e. B ) -> A e. U. B )
7	5, 6	mpancom 420	1 |- ( suc A e. B -> A e. U. B )

Syntax hints:   -> wi 4   e. wcel 2141   _Vcvv 2730   U.cuni 3796   suc csuc 4350

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-uni 3797  df-suc 4356

This theorem is referenced by:  nnsucuniel  6474