Theorem bnj216 34747

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 6465	. 2 ⊢ 𝐵 ∈ suc 𝐵
3		eleq2 2829	. 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 1539   ∈ wcel 2107  Vcvv 3479  suc csuc 6385

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-v 3481  df-un 3955  df-sn 4626  df-suc 6389

This theorem is referenced by:  bnj219  34748  bnj1098  34798  bnj556  34915  bnj557  34916  bnj594  34927  bnj944  34953  bnj966  34959  bnj969  34961  bnj1145  35008