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Theorem bnj216 30054
 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 5721 . 2 𝐵 ∈ suc 𝐵
3 eleq2 2677 . 2 (𝐴 = suc 𝐵 → (𝐵𝐴𝐵 ∈ suc 𝐵))
42, 3mpbiri 247 1 (𝐴 = suc 𝐵𝐵𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173  suc csuc 5642 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-un 3545  df-sn 4126  df-suc 5646 This theorem is referenced by:  bnj219  30055  bnj1098  30108  bnj556  30224  bnj557  30225  bnj594  30236  bnj944  30262  bnj966  30268  bnj969  30270  bnj1145  30315
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