Theorem elint 3933

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 2413	. . . 4 ⊢ (y = A → (y ∈ x ↔ A ∈ x))
3	2	imbi2d 307	. . 3 ⊢ (y = A → ((x ∈ B → y ∈ x) ↔ (x ∈ B → A ∈ x)))
4	3	albidv 1625	. 2 ⊢ (y = A → (∀x(x ∈ B → y ∈ x) ↔ ∀x(x ∈ B → A ∈ x)))
5		df-int 3928	. 2 ⊢ ∩B = {y ∣ ∀x(x ∈ B → y ∈ x)}
6	1, 4, 5	elab2 2989	1 ⊢ (A ∈ ∩B ↔ ∀x(x ∈ B → A ∈ x))

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540   = wceq 1642   ∈ wcel 1710  Vcvv 2860  ∩cint 3927

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-int 3928

This theorem is referenced by:  elint2  3934  elintab  3938  intss1  3942  intss  3948  intun  3959  intpr  3960