Theorem elriin 3748

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 3155	. 2 ⊢ (𝐵 ∈ (𝐴 ∩ ∩ 𝑥 ∈ 𝑋 𝑆) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 ∈ ∩ 𝑥 ∈ 𝑋 𝑆))
2		eliin 3683	. . 3 ⊢ (𝐵 ∈ 𝐴 → (𝐵 ∈ ∩ 𝑥 ∈ 𝑋 𝑆 ↔ ∀𝑥 ∈ 𝑋 𝐵 ∈ 𝑆))
3	2	pm5.32i 441	. 2 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐵 ∈ ∩ 𝑥 ∈ 𝑋 𝑆) ↔ (𝐵 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝑋 𝐵 ∈ 𝑆))
4	1, 3	bitri 182	1 ⊢ (𝐵 ∈ (𝐴 ∩ ∩ 𝑥 ∈ 𝑋 𝑆) ↔ (𝐵 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝑋 𝐵 ∈ 𝑆))

Syntax hints:   ∧ wa 102   ↔ wb 103   ∈ wcel 1433  ∀wral 2348   ∩ cin 2972  ∩ ciin 3679

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-v 2603  df-in 2979  df-iin 3681

This theorem is referenced by: (None)