Theorem notrab 3533

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 3524	. 2 ⊢ ({x ∣ x ∈ A} ∖ {x ∣ φ}) = {x ∣ (x ∈ A ∧ ¬ φ)}
2		difin 3493	. . 3 ⊢ (A ∖ (A ∩ {x ∣ φ})) = (A ∖ {x ∣ φ})
3		dfrab3 3532	. . . 4 ⊢ {x ∈ A ∣ φ} = (A ∩ {x ∣ φ})
4	3	difeq2i 3383	. . 3 ⊢ (A ∖ {x ∈ A ∣ φ}) = (A ∖ (A ∩ {x ∣ φ}))
5		abid2 2471	. . . 4 ⊢ {x ∣ x ∈ A} = A
6	5	difeq1i 3382	. . 3 ⊢ ({x ∣ x ∈ A} ∖ {x ∣ φ}) = (A ∖ {x ∣ φ})
7	2, 4, 6	3eqtr4i 2383	. 2 ⊢ (A ∖ {x ∈ A ∣ φ}) = ({x ∣ x ∈ A} ∖ {x ∣ φ})
8		df-rab 2624	. 2 ⊢ {x ∈ A ∣ ¬ φ} = {x ∣ (x ∈ A ∧ ¬ φ)}
9	1, 7, 8	3eqtr4i 2383	1 ⊢ (A ∖ {x ∈ A ∣ φ}) = {x ∈ A ∣ ¬ φ}

Syntax hints:  ¬ wn 3   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {cab 2339  {crab 2619   ∖ cdif 3207   ∩ cin 3209

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-rab 2624  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216

This theorem is referenced by: (None)