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Theorem intiin 5019
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 4909 . 2 𝐴 = {𝑦 ∣ ∀𝑥𝐴 𝑦𝑥}
2 df-iin 4954 . 2 𝑥𝐴 𝑥 = {𝑦 ∣ ∀𝑥𝐴 𝑦𝑥}
31, 2eqtr4i 2791 1 𝐴 = 𝑥𝐴 𝑥
Colors of variables: wff setvar class
Syntax hints:   = wceq 1563  {cab 2743  wral 3079   cint 4907   ciin 4952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-9 2155  ax-ext 2737
This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-ral 3080  df-int 4908  df-iin 4954
This theorem is referenced by:  trint  5229  relint  5796  intpreima  7055  ixpint  8911  firest  17473  efger  19776  subdrgint  20872  rintopn  23023  intcld  23154  iundifdifd  32812  iundifdif  32813  intxp  49462
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