Theorem elrint 3968

Description: Membership in a restricted intersection. (Contributed by Stefan O'Rear, 3-Apr-2015.)

Assertion

Ref	Expression
elrint	⊢ (X ∈ (A ∩ ∩B) ↔ (X ∈ A ∧ ∀y ∈ B X ∈ y))

Distinct variable groups:   y,B   y,X

Allowed substitution hint:   A(y)

Proof of Theorem elrint

Step	Hyp	Ref	Expression
1		elin 3220	. 2 ⊢ (X ∈ (A ∩ ∩B) ↔ (X ∈ A ∧ X ∈ ∩B))
2		elintg 3935	. . 3 ⊢ (X ∈ A → (X ∈ ∩B ↔ ∀y ∈ B X ∈ y))
3	2	pm5.32i 618	. 2 ⊢ ((X ∈ A ∧ X ∈ ∩B) ↔ (X ∈ A ∧ ∀y ∈ B X ∈ y))
4	1, 3	bitri 240	1 ⊢ (X ∈ (A ∩ ∩B) ↔ (X ∈ A ∧ ∀y ∈ B X ∈ y))

Syntax hints:   ↔ wb 176   ∧ wa 358   ∈ wcel 1710  ∀wral 2615   ∩ cin 3209  ∩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-nan 1288  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-ral 2620  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-int 3928

This theorem is referenced by:  elrint2  3969