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Theorem intiin 5062
Description: Class intersection in terms of indexed intersection. Definition in [Stoll] p. 44. (Contributed by NM, 28-Jun-1998.)
Assertion
Ref Expression
intiin 𝐴 = 𝑥𝐴 𝑥
Distinct variable group:   𝑥,𝐴

Proof of Theorem intiin
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dfint2 4951 . 2 𝐴 = {𝑦 ∣ ∀𝑥𝐴 𝑦𝑥}
2 df-iin 4999 . 2 𝑥𝐴 𝑥 = {𝑦 ∣ ∀𝑥𝐴 𝑦𝑥}
31, 2eqtr4i 2759 1 𝐴 = 𝑥𝐴 𝑥
Colors of variables: wff setvar class
Syntax hints:   = wceq 1534  {cab 2705  wral 3058   cint 4949   ciin 4997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-ral 3059  df-int 4950  df-iin 4999
This theorem is referenced by:  trint  5283  relint  5821  intpreima  7079  ixpint  8944  firest  17414  efger  19673  subdrgint  20691  rintopn  22824  intcld  22957  iundifdifd  32365  iundifdif  32366
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