Theorem clel4 2939

Description: An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.)

Hypothesis

Ref	Expression
clel4.1	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
clel4	⊢ (𝐴 ∈ 𝐵 ↔ ∀𝑥(𝑥 = 𝐵 → 𝐴 ∈ 𝑥))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem clel4

Step	Hyp	Ref	Expression
1		clel4.1	. . 3 ⊢ 𝐵 ∈ V
2		eleq2 2293	. . 3 ⊢ (𝑥 = 𝐵 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐵))
3	1, 2	ceqsalv 2830	. 2 ⊢ (∀𝑥(𝑥 = 𝐵 → 𝐴 ∈ 𝑥) ↔ 𝐴 ∈ 𝐵)
4	3	bicomi 132	1 ⊢ (𝐴 ∈ 𝐵 ↔ ∀𝑥(𝑥 = 𝐵 → 𝐴 ∈ 𝑥))

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1393   = wceq 1395   ∈ wcel 2200  Vcvv 2799

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-v 2801

This theorem is referenced by:  intpr  3954