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Theorem dfdif2 3544
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 3538 . 2 (𝐴𝐵) = {𝑥 ∣ (𝑥𝐴 ∧ ¬ 𝑥𝐵)}
2 df-rab 2900 . 2 {𝑥𝐴 ∣ ¬ 𝑥𝐵} = {𝑥 ∣ (𝑥𝐴 ∧ ¬ 𝑥𝐵)}
31, 2eqtr4i 2630 1 (𝐴𝐵) = {𝑥𝐴 ∣ ¬ 𝑥𝐵}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 382   = wceq 1474  wcel 1975  {cab 2591  {crab 2895  cdif 3532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-ext 2585
This theorem depends on definitions:  df-bi 195  df-cleq 2598  df-rab 2900  df-dif 3538
This theorem is referenced by:  difeq1  3678  difeq2  3679  nfdif  3688  difidALT  3898  ordintdif  5673  kmlem3  8830  incexc2  14351  cnambfre  32427
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