Theorem bnj219 35092

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 3466	. . 3 ⊢ 𝑚 ∈ V
2	1	bnj216 35091	. 2 ⊢ (𝑛 = suc 𝑚 → 𝑚 ∈ 𝑛)
3		epel 5568	. 2 ⊢ (𝑚 E 𝑛 ↔ 𝑚 ∈ 𝑛)
4	2, 3	sylibr 237	1 ⊢ (𝑛 = suc 𝑚 → 𝑚 E 𝑛)

Syntax hints:   → wi 4   = wceq 1568   class class class wbr 5114   E cep 5564  suc csuc 6366

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-ext 2742  ax-sep 5262  ax-pr 5408

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2099  df-clab 2749  df-cleq 2762  df-clel 2845  df-ne 2966  df-rab 3424  df-v 3464  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-eprel 5565  df-suc 6370

This theorem is referenced by:  bnj605  35265  bnj594  35270  bnj607  35274  bnj1110  35340