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Theorem notrab 4052
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 4044 . 2 ({𝑥𝑥𝐴} ∖ {𝑥𝜑}) = {𝑥 ∣ (𝑥𝐴 ∧ ¬ 𝜑)}
2 difin 4010 . . 3 (𝐴 ∖ (𝐴 ∩ {𝑥𝜑})) = (𝐴 ∖ {𝑥𝜑})
3 dfrab3 4050 . . . 4 {𝑥𝐴𝜑} = (𝐴 ∩ {𝑥𝜑})
43difeq2i 3876 . . 3 (𝐴 ∖ {𝑥𝐴𝜑}) = (𝐴 ∖ (𝐴 ∩ {𝑥𝜑}))
5 abid2 2894 . . . 4 {𝑥𝑥𝐴} = 𝐴
65difeq1i 3875 . . 3 ({𝑥𝑥𝐴} ∖ {𝑥𝜑}) = (𝐴 ∖ {𝑥𝜑})
72, 4, 63eqtr4i 2803 . 2 (𝐴 ∖ {𝑥𝐴𝜑}) = ({𝑥𝑥𝐴} ∖ {𝑥𝜑})
8 df-rab 3070 . 2 {𝑥𝐴 ∣ ¬ 𝜑} = {𝑥 ∣ (𝑥𝐴 ∧ ¬ 𝜑)}
91, 7, 83eqtr4i 2803 1 (𝐴 ∖ {𝑥𝐴𝜑}) = {𝑥𝐴 ∣ ¬ 𝜑}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 382   = wceq 1631  wcel 2145  {cab 2757  {crab 3065  cdif 3720  cin 3722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rab 3070  df-v 3353  df-dif 3726  df-in 3730
This theorem is referenced by:  rlimrege0  14518  ordtcld1  21222  ordtcld2  21223  lhop1lem  23996  rpvmasumlem  25397  finsumvtxdg2ssteplem1  26676  frgrwopreglem3  27496  hasheuni  30487  braew  30645
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