Theorem clel2 3615

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

Hypothesis

Ref	Expression
clel2.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
clel2	⊢ (𝐴 ∈ 𝐵 ↔ ∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem clel2

Step	Hyp	Ref	Expression
1		clel2.1	. 2 ⊢ 𝐴 ∈ V
2		clel2g 3614	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝐵 ↔ ∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵)))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∈ 𝐵 ↔ ∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1539   = wceq 1541   ∈ wcel 2111  Vcvv 3436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806

This theorem is referenced by:  mptelee  28871