Theorem bnj216 34725

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 6468	. 2 ⊢ 𝐵 ∈ suc 𝐵
3		eleq2 2828	. 2 ⊢ (𝐴 = suc 𝐵 → (𝐵 ∈ 𝐴 ↔ 𝐵 ∈ suc 𝐵))
4	2, 3	mpbiri 258	1 ⊢ (𝐴 = suc 𝐵 → 𝐵 ∈ 𝐴)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2106  Vcvv 3478  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

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-un 3968  df-sn 4632  df-suc 6392

This theorem is referenced by:  bnj219  34726  bnj1098  34776  bnj556  34893  bnj557  34894  bnj594  34905  bnj944  34931  bnj966  34937  bnj969  34939  bnj1145  34986