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Theorem bnj216 29076
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 4487 . 2  |-  B  e. 
suc  B
3 eleq2 2357 . 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 1632    e. wcel 1696   _Vcvv 2801   suc csuc 4410
This theorem is referenced by:  bnj219  29077  bnj1098  29131  bnj556  29248  bnj557  29249  bnj594  29260  bnj944  29286  bnj966  29292  bnj969  29294  bnj1145  29339
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-un 3170  df-sn 3659  df-suc 4414
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