Theorem notab 4193

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 3114	. . 3 ⊢ {𝑥 ∈ V ∣ ¬ 𝜑} = {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝜑)}
2		rabab 3466	. . 3 ⊢ {𝑥 ∈ V ∣ ¬ 𝜑} = {𝑥 ∣ ¬ 𝜑}
3	1, 2	eqtr3i 2821	. 2 ⊢ {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝜑)} = {𝑥 ∣ ¬ 𝜑}
4		difab 4192	. . 3 ⊢ ({𝑥 ∣ 𝑥 ∈ V} ∖ {𝑥 ∣ 𝜑}) = {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝜑)}
5		abid2 2926	. . . 4 ⊢ {𝑥 ∣ 𝑥 ∈ V} = V
6	5	difeq1i 4016	. . 3 ⊢ ({𝑥 ∣ 𝑥 ∈ V} ∖ {𝑥 ∣ 𝜑}) = (V ∖ {𝑥 ∣ 𝜑})
7	4, 6	eqtr3i 2821	. 2 ⊢ {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝜑)} = (V ∖ {𝑥 ∣ 𝜑})
8	3, 7	eqtr3i 2821	1 ⊢ {𝑥 ∣ ¬ 𝜑} = (V ∖ {𝑥 ∣ 𝜑})

Syntax hints:  ¬ wn 3   ∧ wa 396   = wceq 1522   ∈ wcel 2081  {cab 2775  {crab 3109  Vcvv 3437   ∖ cdif 3856

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-rab 3114  df-v 3439  df-dif 3862

This theorem is referenced by:  dfif3  4395