Theorem bnj219 34726

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 3482	. . 3 ⊢ 𝑚 ∈ V
2	1	bnj216 34725	. 2 ⊢ (𝑛 = suc 𝑚 → 𝑚 ∈ 𝑛)
3		epel 5592	. 2 ⊢ (𝑚 E 𝑛 ↔ 𝑚 ∈ 𝑛)
4	2, 3	sylibr 234	1 ⊢ (𝑛 = suc 𝑚 → 𝑚 E 𝑛)

Syntax hints:   → wi 4   = wceq 1537   class class class wbr 5148   E cep 5588  suc csuc 6388

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-eprel 5589  df-suc 6392

This theorem is referenced by:  bnj605  34900  bnj594  34905  bnj607  34909  bnj1110  34975