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Theorem dfdif2 3868
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 3862 . 2 (𝐴𝐵) = {𝑥 ∣ (𝑥𝐴 ∧ ¬ 𝑥𝐵)}
2 df-rab 3114 . 2 {𝑥𝐴 ∣ ¬ 𝑥𝐵} = {𝑥 ∣ (𝑥𝐴 ∧ ¬ 𝑥𝐵)}
31, 2eqtr4i 2822 1 (𝐴𝐵) = {𝑥𝐴 ∣ ¬ 𝑥𝐵}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 396   = wceq 1522  wcel 2081  {cab 2775  {crab 3109  cdif 3856
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-9 2091  ax-ext 2769
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1762  df-cleq 2788  df-rab 3114  df-dif 3862
This theorem is referenced by:  dfdif3  4012  difeq1  4013  difeq2  4014  nfdif  4023  difidALT  4251  ordintdif  6115  kmlem3  9424  incexc2  15026  cnambfre  34490  alephiso3  39422
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