Theorem clel4 3626

Description: Alternate definition of membership in 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	. 2 ⊢ 𝐵 ∈ V
2		clel4g 3625	. 2 ⊢ (𝐵 ∈ V → (𝐴 ∈ 𝐵 ↔ ∀𝑥(𝑥 = 𝐵 → 𝐴 ∈ 𝑥)))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∈ 𝐵 ↔ ∀𝑥(𝑥 = 𝐵 → 𝐴 ∈ 𝑥))

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1561   = wceq 1563   ∈ wcel 2145  Vcvv 3457

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1566  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840

This theorem is referenced by: (None)