Theorem elintg 1973

Description: Membership in class intersection, with the sethood requirement expressed as an antecedent.

Assertion

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

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

Proof of Theorem elintg

Step	Hyp	Ref	Expression
1		eleq1 1149	. . 3 ⊢ (y = A → (y ∈ ∩B ↔ A ∈ ∩B))
2		eleq1 1149	. . . 4 ⊢ (y = A → (y ∈ x ↔ A ∈ x))
3	2	biraldv 1219	. . 3 ⊢ (y = A → (∀x ∈ B y ∈ x ↔ ∀x ∈ B A ∈ x))
4	1, 3	bibi12d 477	. 2 ⊢ (y = A → ((y ∈ ∩B ↔ ∀x ∈ B y ∈ x) ↔ (A ∈ ∩B ↔ ∀x ∈ B A ∈ x)))
5		visset 1350	. . 3 ⊢ y ∈ V
6	5	elint2 1972	. 2 ⊢ (y ∈ ∩B ↔ ∀x ∈ B y ∈ x)
7	4, 6	vtoclg 1383	1 ⊢ (A ∈ C → (A ∈ ∩B ↔ ∀x ∈ B A ∈ x))

Syntax hints:   → wi 2   ↔ wb 127   ∈ wel 803   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∩cint 1965

This theorem is referenced by:  onmindif 2312  onmindif2 2313

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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