Theorem bnj216 35130

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypothesis

Ref	Expression
bnj216.1	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
bnj216	⊢ (𝐴 = suc 𝐵 → 𝐵 ∈ 𝐴)

Proof of Theorem bnj216

Step	Hyp	Ref	Expression
1		bnj216.1	. . 3 ⊢ 𝐵 ∈ V
2	1	sucid 6445	. 2 ⊢ 𝐵 ∈ suc 𝐵
3		eleq2 2851	. 2 ⊢ (𝐴 = suc 𝐵 → (𝐵 ∈ 𝐴 ↔ 𝐵 ∈ suc 𝐵))
4	2, 3	mpbiri 261	1 ⊢ (𝐴 = suc 𝐵 → 𝐵 ∈ 𝐴)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2142  Vcvv 3454  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

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3456  df-un 3909  df-sn 4589  df-suc 6366

This theorem is used by:  bnj219  35131  bnj1098  35181  bnj556  35297  bnj557  35298  bnj594  35309  bnj944  35335  bnj966  35341  bnj969  35343  bnj1145  35390