Theorem elriin 3972

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 3333	. 2 ⊢ (𝐵 ∈ (𝐴 ∩ ∩ 𝑥 ∈ 𝑋 𝑆) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 ∈ ∩ 𝑥 ∈ 𝑋 𝑆))
2		eliin 3906	. . 3 ⊢ (𝐵 ∈ 𝐴 → (𝐵 ∈ ∩ 𝑥 ∈ 𝑋 𝑆 ↔ ∀𝑥 ∈ 𝑋 𝐵 ∈ 𝑆))
3	2	pm5.32i 454	. 2 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐵 ∈ ∩ 𝑥 ∈ 𝑋 𝑆) ↔ (𝐵 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝑋 𝐵 ∈ 𝑆))
4	1, 3	bitri 184	1 ⊢ (𝐵 ∈ (𝐴 ∩ ∩ 𝑥 ∈ 𝑋 𝑆) ↔ (𝐵 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝑋 𝐵 ∈ 𝑆))

Syntax hints:   ∧ wa 104   ↔ wb 105   ∈ wcel 2160  ∀wral 2468   ∩ cin 3143  ∩ ciin 3902

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-v 2754  df-in 3150  df-iin 3904

This theorem is referenced by: (None)