Theorem compab 37666

Description: Two ways of saying "the complement of a class abstraction". (Contributed by Andrew Salmon, 15-Jul-2011.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)

Assertion

Ref	Expression
compab	⊢ (V ∖ {𝑧 ∣ 𝜑}) = {𝑧 ∣ ¬ 𝜑}

Proof of Theorem compab

Step	Hyp	Ref	Expression
1		nfcv 2751	. . . 4 ⊢ Ⅎ𝑧V
2		nfab1 2753	. . . 4 ⊢ Ⅎ𝑧{𝑧 ∣ 𝜑}
3	1, 2	nfdif 3693	. . 3 ⊢ Ⅎ𝑧(V ∖ {𝑧 ∣ 𝜑})
4		nfab1 2753	. . 3 ⊢ Ⅎ𝑧{𝑧 ∣ ¬ 𝜑}
5	3, 4	cleqf 2776	. 2 ⊢ ((V ∖ {𝑧 ∣ 𝜑}) = {𝑧 ∣ ¬ 𝜑} ↔ ∀𝑧(𝑧 ∈ (V ∖ {𝑧 ∣ 𝜑}) ↔ 𝑧 ∈ {𝑧 ∣ ¬ 𝜑}))
6		abid 2598	. . . 4 ⊢ (𝑧 ∈ {𝑧 ∣ 𝜑} ↔ 𝜑)
7	6	notbii 309	. . 3 ⊢ (¬ 𝑧 ∈ {𝑧 ∣ 𝜑} ↔ ¬ 𝜑)
8		compel 37663	. . 3 ⊢ (𝑧 ∈ (V ∖ {𝑧 ∣ 𝜑}) ↔ ¬ 𝑧 ∈ {𝑧 ∣ 𝜑})
9		abid 2598	. . 3 ⊢ (𝑧 ∈ {𝑧 ∣ ¬ 𝜑} ↔ ¬ 𝜑)
10	7, 8, 9	3bitr4i 291	. 2 ⊢ (𝑧 ∈ (V ∖ {𝑧 ∣ 𝜑}) ↔ 𝑧 ∈ {𝑧 ∣ ¬ 𝜑})
11	5, 10	mpgbir 1717	1 ⊢ (V ∖ {𝑧 ∣ 𝜑}) = {𝑧 ∣ ¬ 𝜑}

Syntax hints:  ¬ wn 3   ↔ wb 195   = wceq 1475   ∈ wcel 1977  {cab 2596  Vcvv 3173   ∖ cdif 3537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543

This theorem is referenced by: (None)