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Theorem notrab 4245
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 4234 . 2 ({𝑥𝑥𝐴} ∖ {𝑥𝜑}) = {𝑥 ∣ (𝑥𝐴 ∧ ¬ 𝜑)}
2 difin 4195 . . 3 (𝐴 ∖ (𝐴 ∩ {𝑥𝜑})) = (𝐴 ∖ {𝑥𝜑})
3 dfrab3 4243 . . . 4 {𝑥𝐴𝜑} = (𝐴 ∩ {𝑥𝜑})
43difeq2i 4054 . . 3 (𝐴 ∖ {𝑥𝐴𝜑}) = (𝐴 ∖ (𝐴 ∩ {𝑥𝜑}))
5 abid2 2882 . . . 4 {𝑥𝑥𝐴} = 𝐴
65difeq1i 4053 . . 3 ({𝑥𝑥𝐴} ∖ {𝑥𝜑}) = (𝐴 ∖ {𝑥𝜑})
72, 4, 63eqtr4i 2776 . 2 (𝐴 ∖ {𝑥𝐴𝜑}) = ({𝑥𝑥𝐴} ∖ {𝑥𝜑})
8 df-rab 3073 . 2 {𝑥𝐴 ∣ ¬ 𝜑} = {𝑥 ∣ (𝑥𝐴 ∧ ¬ 𝜑)}
91, 7, 83eqtr4i 2776 1 (𝐴 ∖ {𝑥𝐴𝜑}) = {𝑥𝐴 ∣ ¬ 𝜑}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 396   = wceq 1539  wcel 2106  {cab 2715  {crab 3068  cdif 3884  cin 3886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-in 3894
This theorem is referenced by:  rlimrege0  15288  mhpmulcl  21339  ordtcld1  22348  ordtcld2  22349  lhop1lem  25177  rpvmasumlem  26635  finsumvtxdg2ssteplem1  27912  frgrwopreglem3  28678  zarcls  31824  hasheuni  32053  braew  32210  satfv1  33325
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