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Theorem dfdif2 3729
Description: Alternate definition of class difference. (Contributed by NM, 25-Mar-2004.)
Assertion
Ref Expression
dfdif2 (𝐴𝐵) = {𝑥𝐴 ∣ ¬ 𝑥𝐵}
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem dfdif2
StepHypRef Expression
1 df-dif 3723 . 2 (𝐴𝐵) = {𝑥 ∣ (𝑥𝐴 ∧ ¬ 𝑥𝐵)}
2 df-rab 3068 . 2 {𝑥𝐴 ∣ ¬ 𝑥𝐵} = {𝑥 ∣ (𝑥𝐴 ∧ ¬ 𝑥𝐵)}
31, 2eqtr4i 2794 1 (𝐴𝐵) = {𝑥𝐴 ∣ ¬ 𝑥𝐵}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 383   = wceq 1629  wcel 2143  {cab 2755  {crab 3063  cdif 3717
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1868  ax-4 1883  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2152  ax-ext 2749
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1851  df-cleq 2762  df-rab 3068  df-dif 3723
This theorem is referenced by:  dfdif3  3868  difeq1  3869  difeq2  3870  nfdif  3879  difidALT  4093  ordintdif  5916  kmlem3  9174  incexc2  14781  cnambfre  33790
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