Theorem elint 4844

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 variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elint.1	. 2 ⊢ 𝐴 ∈ V
2		eleq1 2877	. . . . 5 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ 𝑥 ↔ 𝑦 ∈ 𝑥))
3	2	imbi2d 344	. . . 4 ⊢ (𝑧 = 𝑦 → ((𝑥 ∈ 𝐵 → 𝑧 ∈ 𝑥) ↔ (𝑥 ∈ 𝐵 → 𝑦 ∈ 𝑥)))
4	3	albidv 1921	. . 3 ⊢ (𝑧 = 𝑦 → (∀𝑥(𝑥 ∈ 𝐵 → 𝑧 ∈ 𝑥) ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝑦 ∈ 𝑥)))
5		eleq1 2877	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥))
6	5	imbi2d 344	. . . 4 ⊢ (𝑦 = 𝐴 → ((𝑥 ∈ 𝐵 → 𝑦 ∈ 𝑥) ↔ (𝑥 ∈ 𝐵 → 𝐴 ∈ 𝑥)))
7	6	albidv 1921	. . 3 ⊢ (𝑦 = 𝐴 → (∀𝑥(𝑥 ∈ 𝐵 → 𝑦 ∈ 𝑥) ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝐴 ∈ 𝑥)))
8		df-int 4839	. . 3 ⊢ ∩ 𝐵 = {𝑧 ∣ ∀𝑥(𝑥 ∈ 𝐵 → 𝑧 ∈ 𝑥)}
9	4, 7, 8	elab2gw 3613	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ ∩ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝐴 ∈ 𝑥)))
10	1, 9	ax-mp 5	1 ⊢ (𝐴 ∈ ∩ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝐴 ∈ 𝑥))

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536   = wceq 1538   ∈ wcel 2111  Vcvv 3441  ∩ cint 4838

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-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-int 4839

This theorem is referenced by:  elint2  4845  elintab  4849  intss1  4853  intun  4870  intpr  4871  cssmre  20382  elintfv  33120  dfom5b  33486