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Theorem bnj219 31139
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 3354 . . 3 𝑚 ∈ V
21bnj216 31138 . 2 (𝑛 = suc 𝑚𝑚𝑛)
3 epel 5166 . 2 (𝑚 E 𝑛𝑚𝑛)
42, 3sylibr 224 1 (𝑛 = suc 𝑚𝑚 E 𝑛)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631   class class class wbr 4787   E cep 5162  suc csuc 5867
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-br 4788  df-opab 4848  df-eprel 5163  df-suc 5871
This theorem is referenced by:  bnj605  31315  bnj594  31320  bnj607  31324  bnj1110  31388
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