Theorem elint 4882

Description: Membership in class intersection. (Contributed by NM, 21-May-1994.)

Hypothesis

Ref	Expression
elint.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
elint	⊢ (𝐴 ∈ ∩ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝐴 ∈ 𝑥))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem elint

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elint.1	. 2 ⊢ 𝐴 ∈ V
2		eleq1 2826	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥))
3	2	imbi2d 340	. . . 4 ⊢ (𝑦 = 𝐴 → ((𝑥 ∈ 𝐵 → 𝑦 ∈ 𝑥) ↔ (𝑥 ∈ 𝐵 → 𝐴 ∈ 𝑥)))
4	3	albidv 1924	. . 3 ⊢ (𝑦 = 𝐴 → (∀𝑥(𝑥 ∈ 𝐵 → 𝑦 ∈ 𝑥) ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝐴 ∈ 𝑥)))
5		df-int 4877	. . 3 ⊢ ∩ 𝐵 = {𝑦 ∣ ∀𝑥(𝑥 ∈ 𝐵 → 𝑦 ∈ 𝑥)}
6	4, 5	elab2g 3604	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ ∩ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝐴 ∈ 𝑥)))
7	1, 6	ax-mp 5	1 ⊢ (𝐴 ∈ ∩ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝐴 ∈ 𝑥))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1537   = wceq 1539   ∈ wcel 2108  Vcvv 3422  ∩ cint 4876

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-int 4877

This theorem is referenced by:  elint2  4883  elintab  4887  intss1  4891  intun  4908  intprg  4909  intprOLD  4911  cssmre  20810  elintfv  33644  dfom5b  34141