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Theorem bnj216 29529
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 5513 . 2  |-  B  e. 
suc  B
3 eleq2 2493 . 2  |-  ( A  =  suc  B  -> 
( B  e.  A  <->  B  e.  suc  B ) )
42, 3mpbiri 236 1  |-  ( A  =  suc  B  ->  B  e.  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1867   _Vcvv 3078   suc csuc 5436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-v 3080  df-un 3438  df-sn 3994  df-suc 5440
This theorem is referenced by:  bnj219  29530  bnj1098  29584  bnj556  29700  bnj557  29701  bnj594  29712  bnj944  29738  bnj966  29744  bnj969  29746  bnj1145  29791
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