Theorem bnj219 32712

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 𝑚 → 𝑚 E 𝑛)

Proof of Theorem bnj219

Step	Hyp	Ref	Expression
1		vex 3436	. . 3 ⊢ 𝑚 ∈ V
2	1	bnj216 32711	. 2 ⊢ (𝑛 = suc 𝑚 → 𝑚 ∈ 𝑛)
3		epel 5498	. 2 ⊢ (𝑚 E 𝑛 ↔ 𝑚 ∈ 𝑛)
4	2, 3	sylibr 233	1 ⊢ (𝑛 = suc 𝑚 → 𝑚 E 𝑛)

Syntax hints:   → wi 4   = wceq 1539   class class class wbr 5074   E cep 5494  suc csuc 6268

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-eprel 5495  df-suc 6272

This theorem is referenced by:  bnj605  32887  bnj594  32892  bnj607  32896  bnj1110  32962