MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elrint2 Structured version   Visualization version   GIF version

Theorem elrint2 4923
Description: Membership in a restricted intersection. (Contributed by Stefan O'Rear, 3-Apr-2015.)
Assertion
Ref Expression
elrint2 (𝑋𝐴 → (𝑋 ∈ (𝐴 𝐵) ↔ ∀𝑦𝐵 𝑋𝑦))
Distinct variable groups:   𝑦,𝐵   𝑦,𝑋
Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem elrint2
StepHypRef Expression
1 elrint 4922 . 2 (𝑋 ∈ (𝐴 𝐵) ↔ (𝑋𝐴 ∧ ∀𝑦𝐵 𝑋𝑦))
21baib 536 1 (𝑋𝐴 → (𝑋 ∈ (𝐴 𝐵) ↔ ∀𝑦𝐵 𝑋𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wcel 2106  wral 3064  cin 3886   cint 4879
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-v 3434  df-in 3894  df-int 4880
This theorem is referenced by:  mreacs  17367
  Copyright terms: Public domain W3C validator