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Theorem bnj216 30774
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 5792 . 2 𝐵 ∈ suc 𝐵
3 eleq2 2688 . 2 (𝐴 = suc 𝐵 → (𝐵𝐴𝐵 ∈ suc 𝐵))
42, 3mpbiri 248 1 (𝐴 = suc 𝐵𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1481  wcel 1988  Vcvv 3195  suc csuc 5713
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-v 3197  df-un 3572  df-sn 4169  df-suc 5717
This theorem is referenced by:  bnj219  30775  bnj1098  30828  bnj556  30944  bnj557  30945  bnj594  30956  bnj944  30982  bnj966  30988  bnj969  30990  bnj1145  31035
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