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Theorem bnj216 32029
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 ∈ V
Assertion
Ref Expression
bnj216 (𝐴 = suc 𝐵𝐵𝐴)

Proof of Theorem bnj216
StepHypRef Expression
1 bnj216.1 . . 3 𝐵 ∈ V
21sucid 6258 . 2 𝐵 ∈ suc 𝐵
3 eleq2 2904 . 2 (𝐴 = suc 𝐵 → (𝐵𝐴𝐵 ∈ suc 𝐵))
42, 3mpbiri 261 1 (𝐴 = suc 𝐵𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  Vcvv 3480  suc csuc 6181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-v 3482  df-un 3924  df-sn 4551  df-suc 6185
This theorem is referenced by:  bnj219  32030  bnj1098  32082  bnj556  32199  bnj557  32200  bnj594  32211  bnj944  32237  bnj966  32243  bnj969  32245  bnj1145  32292
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