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Theorem bnj216 28760
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj216.1  |-  B  e. 
_V
Assertion
Ref Expression
bnj216  |-  ( A  =  suc  B  ->  B  e.  A )

Proof of Theorem bnj216
StepHypRef Expression
1 bnj216.1 . . 3  |-  B  e. 
_V
21sucid 4471 . 2  |-  B  e. 
suc  B
3 eleq2 2344 . 2  |-  ( A  =  suc  B  -> 
( B  e.  A  <->  B  e.  suc  B ) )
42, 3mpbiri 224 1  |-  ( A  =  suc  B  ->  B  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   _Vcvv 2788   suc csuc 4394
This theorem is referenced by:  bnj219  28761  bnj1098  28815  bnj556  28932  bnj557  28933  bnj594  28944  bnj944  28970  bnj966  28976  bnj969  28978  bnj1145  29023
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-un 3157  df-sn 3646  df-suc 4398
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