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Theorem bnj219 34042
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 3476 . . 3 𝑚 ∈ V
21bnj216 34041 . 2 (𝑛 = suc 𝑚𝑚𝑛)
3 epel 5582 . 2 (𝑚 E 𝑛𝑚𝑛)
42, 3sylibr 233 1 (𝑛 = suc 𝑚𝑚 E 𝑛)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539   class class class wbr 5147   E cep 5578  suc csuc 6365
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-eprel 5579  df-suc 6369
This theorem is referenced by:  bnj605  34216  bnj594  34221  bnj607  34225  bnj1110  34291
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