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Theorem elrint 3882
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 3318 . 2 (𝑋 ∈ (𝐴 𝐵) ↔ (𝑋𝐴𝑋 𝐵))
2 elintg 3850 . . 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 2148  wral 2455  cin 3128   cint 3842
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2739  df-in 3135  df-int 3843
This theorem is referenced by:  elrint2  3883
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