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Theorem dfdif2 3124
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 3118 . 2 (𝐴𝐵) = {𝑥 ∣ (𝑥𝐴 ∧ ¬ 𝑥𝐵)}
2 df-rab 2453 . 2 {𝑥𝐴 ∣ ¬ 𝑥𝐵} = {𝑥 ∣ (𝑥𝐴 ∧ ¬ 𝑥𝐵)}
31, 2eqtr4i 2189 1 (𝐴𝐵) = {𝑥𝐴 ∣ ¬ 𝑥𝐵}
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 103   = wceq 1343  wcel 2136  {cab 2151  {crab 2448  cdif 3113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-4 1498  ax-17 1514  ax-ext 2147
This theorem depends on definitions:  df-bi 116  df-cleq 2158  df-rab 2453  df-dif 3118
This theorem is referenced by:  dfdif3  3232  difeq1  3233  difeq2  3234  nfdif  3243  difidALT  3478
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