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Theorem bnj216 29101
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 5488 . 2  |-  B  e. 
suc  B
3 eleq2 2475 . 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 1405    e. wcel 1842   _Vcvv 3058   suc csuc 5411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-v 3060  df-un 3418  df-sn 3972  df-suc 5415
This theorem is referenced by:  bnj219  29102  bnj1098  29156  bnj556  29272  bnj557  29273  bnj594  29284  bnj944  29310  bnj966  29316  bnj969  29318  bnj1145  29363
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