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Theorem notrab 4284
Description: Complementation of restricted class abstractions. (Contributed by Mario Carneiro, 3-Sep-2015.)
Assertion
Ref Expression
notrab (𝐴 ∖ {𝑥𝐴𝜑}) = {𝑥𝐴 ∣ ¬ 𝜑}
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem notrab
StepHypRef Expression
1 difab 4276 . 2 ({𝑥𝑥𝐴} ∖ {𝑥𝜑}) = {𝑥 ∣ (𝑥𝐴 ∧ ¬ 𝜑)}
2 difin 4242 . . 3 (𝐴 ∖ (𝐴 ∩ {𝑥𝜑})) = (𝐴 ∖ {𝑥𝜑})
3 dfrab3 4282 . . . 4 {𝑥𝐴𝜑} = (𝐴 ∩ {𝑥𝜑})
43difeq2i 4100 . . 3 (𝐴 ∖ {𝑥𝐴𝜑}) = (𝐴 ∖ (𝐴 ∩ {𝑥𝜑}))
5 abid2 2962 . . . 4 {𝑥𝑥𝐴} = 𝐴
65difeq1i 4099 . . 3 ({𝑥𝑥𝐴} ∖ {𝑥𝜑}) = (𝐴 ∖ {𝑥𝜑})
72, 4, 63eqtr4i 2859 . 2 (𝐴 ∖ {𝑥𝐴𝜑}) = ({𝑥𝑥𝐴} ∖ {𝑥𝜑})
8 df-rab 3152 . 2 {𝑥𝐴 ∣ ¬ 𝜑} = {𝑥 ∣ (𝑥𝐴 ∧ ¬ 𝜑)}
91, 7, 83eqtr4i 2859 1 (𝐴 ∖ {𝑥𝐴𝜑}) = {𝑥𝐴 ∣ ¬ 𝜑}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 396   = wceq 1530  wcel 2107  {cab 2804  {crab 3147  cdif 3937  cin 3939
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-rab 3152  df-v 3502  df-dif 3943  df-in 3947
This theorem is referenced by:  rlimrege0  14931  ordtcld1  21740  ordtcld2  21741  lhop1lem  24544  rpvmasumlem  25996  finsumvtxdg2ssteplem1  27260  frgrwopreglem3  28026  hasheuni  31249  braew  31406  satfv1  32513
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