Theorem notab 3526

Description: A class builder defined by a negation. (Contributed by FL, 18-Sep-2010.)

Assertion

Ref	Expression
notab	⊢ {x ∣ ¬ φ} = (V ∖ {x ∣ φ})

Proof of Theorem notab

Step	Hyp	Ref	Expression
1		df-rab 2624	. . 3 ⊢ {x ∈ V ∣ ¬ φ} = {x ∣ (x ∈ V ∧ ¬ φ)}
2		rabab 2877	. . 3 ⊢ {x ∈ V ∣ ¬ φ} = {x ∣ ¬ φ}
3	1, 2	eqtr3i 2375	. 2 ⊢ {x ∣ (x ∈ V ∧ ¬ φ)} = {x ∣ ¬ φ}
4		difab 3524	. . 3 ⊢ ({x ∣ x ∈ V} ∖ {x ∣ φ}) = {x ∣ (x ∈ V ∧ ¬ φ)}
5		abid2 2471	. . . 4 ⊢ {x ∣ x ∈ V} = V
6	5	difeq1i 3382	. . 3 ⊢ ({x ∣ x ∈ V} ∖ {x ∣ φ}) = (V ∖ {x ∣ φ})
7	4, 6	eqtr3i 2375	. 2 ⊢ {x ∣ (x ∈ V ∧ ¬ φ)} = (V ∖ {x ∣ φ})
8	3, 7	eqtr3i 2375	1 ⊢ {x ∣ ¬ φ} = (V ∖ {x ∣ φ})

Syntax hints:  ¬ wn 3   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {cab 2339  {crab 2619  Vcvv 2860   ∖ cdif 3207

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:  dfif3  3673