Theorem notrab 4242

Description: Complementation of restricted class abstractions. (Contributed by Mario Carneiro, 3-Sep-2015.)

Assertion

Ref	Expression
notrab	⊢ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝜑}) = {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem notrab

Step	Hyp	Ref	Expression
1		difab 4231	. 2 ⊢ ({𝑥 ∣ 𝑥 ∈ 𝐴} ∖ {𝑥 ∣ 𝜑}) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝜑)}
2		difin 4192	. . 3 ⊢ (𝐴 ∖ (𝐴 ∩ {𝑥 ∣ 𝜑})) = (𝐴 ∖ {𝑥 ∣ 𝜑})
3		dfrab3 4240	. . . 4 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = (𝐴 ∩ {𝑥 ∣ 𝜑})
4	3	difeq2i 4050	. . 3 ⊢ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝜑}) = (𝐴 ∖ (𝐴 ∩ {𝑥 ∣ 𝜑}))
5		abid2 2881	. . . 4 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴
6	5	difeq1i 4049	. . 3 ⊢ ({𝑥 ∣ 𝑥 ∈ 𝐴} ∖ {𝑥 ∣ 𝜑}) = (𝐴 ∖ {𝑥 ∣ 𝜑})
7	2, 4, 6	3eqtr4i 2776	. 2 ⊢ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝜑}) = ({𝑥 ∣ 𝑥 ∈ 𝐴} ∖ {𝑥 ∣ 𝜑})
8		df-rab 3072	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝜑)}
9	1, 7, 8	3eqtr4i 2776	1 ⊢ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝜑}) = {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1539   ∈ wcel 2108  {cab 2715  {crab 3067   ∖ cdif 3880   ∩ cin 3882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-in 3890

This theorem is referenced by:  rlimrege0  15216  mhpmulcl  21249  ordtcld1  22256  ordtcld2  22257  lhop1lem  25082  rpvmasumlem  26540  finsumvtxdg2ssteplem1  27815  frgrwopreglem3  28579  zarcls  31726  hasheuni  31953  braew  32110  satfv1  33225