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Theorem elrint 4994
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 3979 . 2 (𝑋 ∈ (𝐴 𝐵) ↔ (𝑋𝐴𝑋 𝐵))
2 elintg 4959 . . 3 (𝑋𝐴 → (𝑋 𝐵 ↔ ∀𝑦𝐵 𝑋𝑦))
32pm5.32i 574 . 2 ((𝑋𝐴𝑋 𝐵) ↔ (𝑋𝐴 ∧ ∀𝑦𝐵 𝑋𝑦))
41, 3bitri 275 1 (𝑋 ∈ (𝐴 𝐵) ↔ (𝑋𝐴 ∧ ∀𝑦𝐵 𝑋𝑦))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wcel 2106  wral 3059  cin 3962   cint 4951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-v 3480  df-in 3970  df-int 4952
This theorem is referenced by:  elrint2  4995  ptcnplem  23645  tmdgsum2  24120  limciun  25944
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