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Theorem bnj219 32063
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Assertion
Ref Expression
bnj219 (𝑛 = suc 𝑚𝑚 E 𝑛)

Proof of Theorem bnj219
StepHypRef Expression
1 vex 3483 . . 3 𝑚 ∈ V
21bnj216 32062 . 2 (𝑛 = suc 𝑚𝑚𝑛)
3 epel 5456 . 2 (𝑚 E 𝑛𝑚𝑛)
42, 3sylibr 237 1 (𝑛 = suc 𝑚𝑚 E 𝑛)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538   class class class wbr 5052   E cep 5451  suc csuc 6180
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-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-br 5053  df-opab 5115  df-eprel 5452  df-suc 6184
This theorem is referenced by:  bnj605  32239  bnj594  32244  bnj607  32248  bnj1110  32314
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