Theorem bnj219 35131

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 3458	. . 3 ⊢ 𝑚 ∈ V
2	1	bnj216 35130	. 2 ⊢ (𝑛 = suc 𝑚 → 𝑚 ∈ 𝑛)
3		epel 5563	. 2 ⊢ (𝑚 E 𝑛 ↔ 𝑚 ∈ 𝑛)
4	2, 3	sylibr 237	1 ⊢ (𝑛 = suc 𝑚 → 𝑚 E 𝑛)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   class class class wbr 5108   E cep 5559  suc csuc 6362

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5256  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-eprel 5560  df-suc 6366

This theorem is used by:  bnj605  35304  bnj594  35309  bnj607  35313  bnj1110  35379