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Theorem bnj216 32075
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 4909 . 2  |-  B  e. 
suc  B
3 eleq2 2527 . 2  |-  ( A  =  suc  B  -> 
( B  e.  A  <->  B  e.  suc  B ) )
42, 3mpbiri 233 1  |-  ( A  =  suc  B  ->  B  e.  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   _Vcvv 3078   suc csuc 4832
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3080  df-un 3444  df-sn 3989  df-suc 4836
This theorem is referenced by:  bnj219  32076  bnj1098  32129  bnj556  32245  bnj557  32246  bnj594  32257  bnj944  32283  bnj966  32289  bnj969  32291  bnj1145  32336
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