ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  elrint GIF version

Theorem elrint 3902
Description: Membership in a restricted intersection. (Contributed by Stefan O'Rear, 3-Apr-2015.)
Assertion
Ref Expression
elrint (𝑋 ∈ (𝐴 𝐵) ↔ (𝑋𝐴 ∧ ∀𝑦𝐵 𝑋𝑦))
Distinct variable groups:   𝑦,𝐵   𝑦,𝑋
Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem elrint
StepHypRef Expression
1 elin 3333 . 2 (𝑋 ∈ (𝐴 𝐵) ↔ (𝑋𝐴𝑋 𝐵))
2 elintg 3870 . . 3 (𝑋𝐴 → (𝑋 𝐵 ↔ ∀𝑦𝐵 𝑋𝑦))
32pm5.32i 454 . 2 ((𝑋𝐴𝑋 𝐵) ↔ (𝑋𝐴 ∧ ∀𝑦𝐵 𝑋𝑦))
41, 3bitri 184 1 (𝑋 ∈ (𝐴 𝐵) ↔ (𝑋𝐴 ∧ ∀𝑦𝐵 𝑋𝑦))
Colors of variables: wff set class
Syntax hints:  wa 104  wb 105  wcel 2160  wral 2468  cin 3143   cint 3862
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-v 2754  df-in 3150  df-int 3863
This theorem is referenced by:  elrint2  3903
  Copyright terms: Public domain W3C validator