Theorem elint2 1972

Description: Membership in class intersection.

Hypothesis

Ref	Expression
elint2.1	⊢ A ∈ V

Assertion

Ref	Expression
elint2	⊢ (A ∈ ∩B ↔ ∀x ∈ B A ∈ x)

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

Proof of Theorem elint2

Step	Hyp	Ref	Expression
1		elint2.1	. . 3 ⊢ A ∈ V
2	1	elint 1971	. 2 ⊢ (A ∈ ∩B ↔ ∀x(x ∈ B → A ∈ x))
3		df-ral 1205	. 2 ⊢ (∀x ∈ B A ∈ x ↔ ∀x(x ∈ B → A ∈ x))
4	2, 3	bitr4 154	1 ⊢ (A ∈ ∩B ↔ ∀x ∈ B A ∈ x)

Syntax hints:   → wi 2   ↔ wb 127  ∀wal 672   ∈ wcel 1092  ∀wral 1201  Vcvv 1348  ∩cint 1965

This theorem is referenced by:  elintg 1973  ssint 1980  iinuni 2036  onint 2261  shintcl 5294  chintcl 5296

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-ral 1205  df-v 1349  df-int 1966

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