MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elriin Structured version   Visualization version   GIF version

Theorem elriin 4520
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 3754 . 2 (𝐵 ∈ (𝐴 𝑥𝑋 𝑆) ↔ (𝐵𝐴𝐵 𝑥𝑋 𝑆))
2 eliin 4452 . . 3 (𝐵𝐴 → (𝐵 𝑥𝑋 𝑆 ↔ ∀𝑥𝑋 𝐵𝑆))
32pm5.32i 666 . 2 ((𝐵𝐴𝐵 𝑥𝑋 𝑆) ↔ (𝐵𝐴 ∧ ∀𝑥𝑋 𝐵𝑆))
41, 3bitri 262 1 (𝐵 ∈ (𝐴 𝑥𝑋 𝑆) ↔ (𝐵𝐴 ∧ ∀𝑥𝑋 𝐵𝑆))
Colors of variables: wff setvar class
Syntax hints:  wb 194  wa 382  wcel 1976  wral 2892  cin 3535   ciin 4447
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ral 2897  df-v 3171  df-in 3543  df-iin 4449
This theorem is referenced by:  limciun  23378  limcun  23379
  Copyright terms: Public domain W3C validator