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Theorem bnj216 34747
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 6465 . 2 𝐵 ∈ suc 𝐵
3 eleq2 2829 . 2 (𝐴 = suc 𝐵 → (𝐵𝐴𝐵 ∈ suc 𝐵))
42, 3mpbiri 258 1 (𝐴 = suc 𝐵𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2107  Vcvv 3479  suc csuc 6385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-v 3481  df-un 3955  df-sn 4626  df-suc 6389
This theorem is referenced by:  bnj219  34748  bnj1098  34798  bnj556  34915  bnj557  34916  bnj594  34927  bnj944  34953  bnj966  34959  bnj969  34961  bnj1145  35008
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