Theorem bnj219 32113

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 3444	. . 3 ⊢ 𝑚 ∈ V
2	1	bnj216 32112	. 2 ⊢ (𝑛 = suc 𝑚 → 𝑚 ∈ 𝑛)
3		epel 5433	. 2 ⊢ (𝑚 E 𝑛 ↔ 𝑚 ∈ 𝑛)
4	2, 3	sylibr 237	1 ⊢ (𝑛 = suc 𝑚 → 𝑚 E 𝑛)

Syntax hints:   → wi 4   = wceq 1538   class class class wbr 5030   E cep 5429  suc csuc 6161

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-eprel 5430  df-suc 6165

This theorem is referenced by:  bnj605  32289  bnj594  32294  bnj607  32298  bnj1110  32364