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Theorem notrab 4200
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 4190 . 2 ({𝑥𝑥𝐴} ∖ {𝑥𝜑}) = {𝑥 ∣ (𝑥𝐴 ∧ ¬ 𝜑)}
2 difin 4152 . . 3 (𝐴 ∖ (𝐴 ∩ {𝑥𝜑})) = (𝐴 ∖ {𝑥𝜑})
3 dfrab3 4198 . . . 4 {𝑥𝐴𝜑} = (𝐴 ∩ {𝑥𝜑})
43difeq2i 4010 . . 3 (𝐴 ∖ {𝑥𝐴𝜑}) = (𝐴 ∖ (𝐴 ∩ {𝑥𝜑}))
5 abid2 2874 . . . 4 {𝑥𝑥𝐴} = 𝐴
65difeq1i 4009 . . 3 ({𝑥𝑥𝐴} ∖ {𝑥𝜑}) = (𝐴 ∖ {𝑥𝜑})
72, 4, 63eqtr4i 2771 . 2 (𝐴 ∖ {𝑥𝐴𝜑}) = ({𝑥𝑥𝐴} ∖ {𝑥𝜑})
8 df-rab 3062 . 2 {𝑥𝐴 ∣ ¬ 𝜑} = {𝑥 ∣ (𝑥𝐴 ∧ ¬ 𝜑)}
91, 7, 83eqtr4i 2771 1 (𝐴 ∖ {𝑥𝐴𝜑}) = {𝑥𝐴 ∣ ¬ 𝜑}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 399   = wceq 1542  wcel 2114  {cab 2716  {crab 3057  cdif 3840  cin 3842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-12 2179  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-ex 1787  df-nf 1791  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-rab 3062  df-v 3400  df-dif 3846  df-in 3850
This theorem is referenced by:  rlimrege0  15028  mhpmulcl  20945  ordtcld1  21950  ordtcld2  21951  lhop1lem  24767  rpvmasumlem  26225  finsumvtxdg2ssteplem1  27489  frgrwopreglem3  28253  zarcls  31398  hasheuni  31625  braew  31782  satfv1  32898
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