Theorem elriin 4035

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 3387	. 2 ⊢ (𝐵 ∈ (𝐴 ∩ ∩ 𝑥 ∈ 𝑋 𝑆) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 ∈ ∩ 𝑥 ∈ 𝑋 𝑆))
2		eliin 3969	. . 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 2200  ∀wral 2508   ∩ cin 3196  ∩ ciin 3965

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-v 2801  df-in 3203  df-iin 3967

This theorem is referenced by: (None)