Theorem notrab 3486

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 3478	. 2 ⊢ ({𝑥 ∣ 𝑥 ∈ 𝐴} ∖ {𝑥 ∣ 𝜑}) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝜑)}
2		difin 3446	. . 3 ⊢ (𝐴 ∖ (𝐴 ∩ {𝑥 ∣ 𝜑})) = (𝐴 ∖ {𝑥 ∣ 𝜑})
3		dfrab3 3485	. . . 4 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = (𝐴 ∩ {𝑥 ∣ 𝜑})
4	3	difeq2i 3324	. . 3 ⊢ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝜑}) = (𝐴 ∖ (𝐴 ∩ {𝑥 ∣ 𝜑}))
5		abid2 2353	. . . 4 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴
6	5	difeq1i 3323	. . 3 ⊢ ({𝑥 ∣ 𝑥 ∈ 𝐴} ∖ {𝑥 ∣ 𝜑}) = (𝐴 ∖ {𝑥 ∣ 𝜑})
7	2, 4, 6	3eqtr4i 2262	. 2 ⊢ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝜑}) = ({𝑥 ∣ 𝑥 ∈ 𝐴} ∖ {𝑥 ∣ 𝜑})
8		df-rab 2520	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝜑)}
9	1, 7, 8	3eqtr4i 2262	1 ⊢ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝜑}) = {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}

Syntax hints:  ¬ wn 3   ∧ wa 104   = wceq 1398   ∈ wcel 2202  {cab 2217  {crab 2515   ∖ cdif 3198   ∩ cin 3200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rab 2520  df-v 2805  df-dif 3203  df-in 3207

This theorem is referenced by:  diffitest  7119