Theorem notab 4294

Description: A class abstraction 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 3420	. . 3 ⊢ {𝑥 ∈ V ∣ ¬ 𝜑} = {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝜑)}
2		rabab 3495	. . 3 ⊢ {𝑥 ∈ V ∣ ¬ 𝜑} = {𝑥 ∣ ¬ 𝜑}
3	1, 2	eqtr3i 2759	. 2 ⊢ {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝜑)} = {𝑥 ∣ ¬ 𝜑}
4		difab 4290	. . 3 ⊢ ({𝑥 ∣ 𝑥 ∈ V} ∖ {𝑥 ∣ 𝜑}) = {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝜑)}
5		abid2 2871	. . . 4 ⊢ {𝑥 ∣ 𝑥 ∈ V} = V
6	5	difeq1i 4102	. . 3 ⊢ ({𝑥 ∣ 𝑥 ∈ V} ∖ {𝑥 ∣ 𝜑}) = (V ∖ {𝑥 ∣ 𝜑})
7	4, 6	eqtr3i 2759	. 2 ⊢ {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝜑)} = (V ∖ {𝑥 ∣ 𝜑})
8	3, 7	eqtr3i 2759	1 ⊢ {𝑥 ∣ ¬ 𝜑} = (V ∖ {𝑥 ∣ 𝜑})

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1539   ∈ wcel 2107  {cab 2712  {crab 3419  Vcvv 3463   ∖ cdif 3928

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-12 2176  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-rab 3420  df-v 3465  df-dif 3934

This theorem is referenced by: (None)