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Theorem intiin 5018
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 4908 . 2 𝐴 = {𝑦 ∣ ∀𝑥𝐴 𝑦𝑥}
2 df-iin 4956 . 2 𝑥𝐴 𝑥 = {𝑦 ∣ ∀𝑥𝐴 𝑦𝑥}
31, 2eqtr4i 2769 1 𝐴 = 𝑥𝐴 𝑥
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  {cab 2715  wral 3063   cint 4906   ciin 4954
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-9 2117  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-sb 2069  df-clab 2716  df-cleq 2730  df-ral 3064  df-int 4907  df-iin 4956
This theorem is referenced by:  trint  5239  relint  5774  intpreima  7018  ixpint  8822  firest  17274  efger  19459  subdrgint  20223  rintopn  22210  intcld  22343  iundifdifd  31308  iundifdif  31309
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