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Theorem elrint 3775
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 3223 . 2 (𝑋 ∈ (𝐴 𝐵) ↔ (𝑋𝐴𝑋 𝐵))
2 elintg 3743 . . 3 (𝑋𝐴 → (𝑋 𝐵 ↔ ∀𝑦𝐵 𝑋𝑦))
32pm5.32i 447 . 2 ((𝑋𝐴𝑋 𝐵) ↔ (𝑋𝐴 ∧ ∀𝑦𝐵 𝑋𝑦))
41, 3bitri 183 1 (𝑋 ∈ (𝐴 𝐵) ↔ (𝑋𝐴 ∧ ∀𝑦𝐵 𝑋𝑦))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104  wcel 1461  wral 2388  cin 3034   cint 3735
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095
This theorem depends on definitions:  df-bi 116  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-v 2657  df-in 3041  df-int 3736
This theorem is referenced by:  elrint2  3776
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