Theorem elriin 4038

Description: Elementhood in a relative intersection. (Contributed by Mario Carneiro, 30-Dec-2016.)

Assertion

Ref	Expression
elriin	⊢ (B ∈ (A ∩ ∩x ∈ X S) ↔ (B ∈ A ∧ ∀x ∈ X B ∈ S))

Distinct variable groups:   x,A   x,X   x,B

Allowed substitution hint:   S(x)

Proof of Theorem elriin

Step	Hyp	Ref	Expression
1		elin 3219	. 2 ⊢ (B ∈ (A ∩ ∩x ∈ X S) ↔ (B ∈ A ∧ B ∈ ∩x ∈ X S))
2		eliin 3974	. . 3 ⊢ (B ∈ A → (B ∈ ∩x ∈ X S ↔ ∀x ∈ X B ∈ S))
3	2	pm5.32i 618	. 2 ⊢ ((B ∈ A ∧ B ∈ ∩x ∈ X S) ↔ (B ∈ A ∧ ∀x ∈ X B ∈ S))
4	1, 3	bitri 240	1 ⊢ (B ∈ (A ∩ ∩x ∈ X S) ↔ (B ∈ A ∧ ∀x ∈ X B ∈ S))

Syntax hints:   ↔ wb 176   ∧ wa 358   ∈ wcel 1710  ∀wral 2614   ∩ cin 3208  ∩ciin 3970

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 2478  df-ral 2619  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-iin 3972

This theorem is referenced by: (None)