Theorem clel4 3604

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 2878	. . 3 ⊢ (𝑥 = 𝐵 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐵))
3	1, 2	ceqsalv 3479	. 2 ⊢ (∀𝑥(𝑥 = 𝐵 → 𝐴 ∈ 𝑥) ↔ 𝐴 ∈ 𝐵)
4	3	bicomi 227	1 ⊢ (𝐴 ∈ 𝐵 ↔ ∀𝑥(𝑥 = 𝐵 → 𝐴 ∈ 𝑥))

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536   = wceq 1538   ∈ wcel 2111  Vcvv 3441

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-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1086  df-ex 1782  df-nf 1786  df-cleq 2791  df-clel 2870

This theorem is referenced by:  intpr  4871