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Theorem sucunielr 4327
Description: Successor and union. The converse (where  B is an ordinal) implies excluded middle, as seen at ordsucunielexmid 4347. (Contributed by Jim Kingdon, 2-Aug-2019.)
Assertion
Ref Expression
sucunielr  |-  ( suc 
A  e.  B  ->  A  e.  U. B )

Proof of Theorem sucunielr
StepHypRef Expression
1 elex 2630 . . . 4  |-  ( suc 
A  e.  B  ->  suc  A  e.  _V )
2 sucexb 4314 . . . 4  |-  ( A  e.  _V  <->  suc  A  e. 
_V )
31, 2sylibr 132 . . 3  |-  ( suc 
A  e.  B  ->  A  e.  _V )
4 sucidg 4243 . . 3  |-  ( A  e.  _V  ->  A  e.  suc  A )
53, 4syl 14 . 2  |-  ( suc 
A  e.  B  ->  A  e.  suc  A )
6 elunii 3658 . 2  |-  ( ( A  e.  suc  A  /\  suc  A  e.  B
)  ->  A  e.  U. B )
75, 6mpancom 413 1  |-  ( suc 
A  e.  B  ->  A  e.  U. B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1438   _Vcvv 2619   U.cuni 3653   suc csuc 4192
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-uni 3654  df-suc 4198
This theorem is referenced by:  nnsucuniel  6256
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