Theorem sucunielr 4254

Description: Successor and union. The converse (where B is an ordinal) implies excluded middle, as seen at ordsucunielexmid 4274. (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 2610	. . . 4 |- ( suc A e. B -> suc A e. _V )
2		sucexb 4241	. . . 4 |- ( A e. _V <-> suc A e. _V )
3	1, 2	sylibr 132	. . 3 |- ( suc A e. B -> A e. _V )
4		sucidg 4171	. . 3 |- ( A e. _V -> A e. suc A )
5	3, 4	syl 14	. 2 |- ( suc A e. B -> A e. suc A )
6		elunii 3606	. 2 |- ( ( A e. suc A /\ suc A e. B ) -> A e. U. B )
7	5, 6	mpancom 413	1 |- ( suc A e. B -> A e. U. B )

Syntax hints:   -> wi 4   e. wcel 1433   _Vcvv 2601   U.cuni 3601   suc csuc 4120

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-uni 3602  df-suc 4126

This theorem is referenced by: (None)