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Theorem elriin 5091
Description: Elementhood in a relative intersection. (Contributed by Mario Carneiro, 30-Dec-2016.)
Assertion
Ref Expression
elriin (𝐵 ∈ (𝐴 𝑥𝑋 𝑆) ↔ (𝐵𝐴 ∧ ∀𝑥𝑋 𝐵𝑆))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑋   𝑥,𝐵
Allowed substitution hint:   𝑆(𝑥)

Proof of Theorem elriin
StepHypRef Expression
1 elin 3963 . 2 (𝐵 ∈ (𝐴 𝑥𝑋 𝑆) ↔ (𝐵𝐴𝐵 𝑥𝑋 𝑆))
2 eliin 5008 . . 3 (𝐵𝐴 → (𝐵 𝑥𝑋 𝑆 ↔ ∀𝑥𝑋 𝐵𝑆))
32pm5.32i 573 . 2 ((𝐵𝐴𝐵 𝑥𝑋 𝑆) ↔ (𝐵𝐴 ∧ ∀𝑥𝑋 𝐵𝑆))
41, 3bitri 274 1 (𝐵 ∈ (𝐴 𝑥𝑋 𝑆) ↔ (𝐵𝐴 ∧ ∀𝑥𝑋 𝐵𝑆))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394  wcel 2099  wral 3051  cin 3946   ciin 5004
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-v 3464  df-in 3954  df-iin 5006
This theorem is referenced by:  limciun  25917  limcun  25918
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