Theorem elriin 5104

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 3992	. 2 ⊢ (𝐵 ∈ (𝐴 ∩ ∩ 𝑥 ∈ 𝑋 𝑆) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 ∈ ∩ 𝑥 ∈ 𝑋 𝑆))
2		eliin 5020	. . 3 ⊢ (𝐵 ∈ 𝐴 → (𝐵 ∈ ∩ 𝑥 ∈ 𝑋 𝑆 ↔ ∀𝑥 ∈ 𝑋 𝐵 ∈ 𝑆))
3	2	pm5.32i 574	. 2 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐵 ∈ ∩ 𝑥 ∈ 𝑋 𝑆) ↔ (𝐵 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝑋 𝐵 ∈ 𝑆))
4	1, 3	bitri 275	1 ⊢ (𝐵 ∈ (𝐴 ∩ ∩ 𝑥 ∈ 𝑋 𝑆) ↔ (𝐵 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝑋 𝐵 ∈ 𝑆))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2108  ∀wral 3067   ∩ cin 3975  ∩ ciin 5016

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-v 3490  df-in 3983  df-iin 5018

This theorem is referenced by:  limciun  25949  limcun  25950