Theorem elint 1971

Description: Membership in class intersection.

Hypothesis

Ref	Expression
elint.1	⊢ A ∈ V

Assertion

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

Distinct variable group(s):   x,A   x,B

Proof of Theorem elint

Step	Hyp	Ref	Expression
1		elint.1	. 2 ⊢ A ∈ V
2		eleq1 1149	. . . 4 ⊢ (y = A → (y ∈ x ↔ A ∈ x))
3	2	imbi2d 464	. . 3 ⊢ (y = A → ((x ∈ B → y ∈ x) ↔ (x ∈ B → A ∈ x)))
4	3	bialdv 935	. 2 ⊢ (y = A → (∀x(x ∈ B → y ∈ x) ↔ ∀x(x ∈ B → A ∈ x)))
5		df-int 1966	. 2 ⊢ ∩B = {y∣∀x(x ∈ B → y ∈ x)}
6	1, 4, 5	elab2 1419	1 ⊢ (A ∈ ∩B ↔ ∀x(x ∈ B → A ∈ x))

Syntax hints:   → wi 2   ↔ wb 127  ∀wal 672   ∈ wel 803   = wceq 1091   ∈ wcel 1092  Vcvv 1348  ∩cint 1965

This theorem is referenced by:  elint2 1972  elinti 1974  hbint 1975  elintab 1976  intss1 1979  intss 1983  intun 1989  intpr 1990

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-int 1966

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