Theorem compab 40767

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 2977	. . . 4 ⊢ Ⅎ𝑧V
2		nfab1 2979	. . . 4 ⊢ Ⅎ𝑧{𝑧 ∣ 𝜑}
3	1, 2	nfdif 4101	. . 3 ⊢ Ⅎ𝑧(V ∖ {𝑧 ∣ 𝜑})
4		nfab1 2979	. . 3 ⊢ Ⅎ𝑧{𝑧 ∣ ¬ 𝜑}
5	3, 4	cleqf 3010	. 2 ⊢ ((V ∖ {𝑧 ∣ 𝜑}) = {𝑧 ∣ ¬ 𝜑} ↔ ∀𝑧(𝑧 ∈ (V ∖ {𝑧 ∣ 𝜑}) ↔ 𝑧 ∈ {𝑧 ∣ ¬ 𝜑}))
6		abid 2803	. . . 4 ⊢ (𝑧 ∈ {𝑧 ∣ 𝜑} ↔ 𝜑)
7	6	notbii 322	. . 3 ⊢ (¬ 𝑧 ∈ {𝑧 ∣ 𝜑} ↔ ¬ 𝜑)
8		velcomp 3950	. . 3 ⊢ (𝑧 ∈ (V ∖ {𝑧 ∣ 𝜑}) ↔ ¬ 𝑧 ∈ {𝑧 ∣ 𝜑})
9		abid 2803	. . 3 ⊢ (𝑧 ∈ {𝑧 ∣ ¬ 𝜑} ↔ ¬ 𝜑)
10	7, 8, 9	3bitr4i 305	. 2 ⊢ (𝑧 ∈ (V ∖ {𝑧 ∣ 𝜑}) ↔ 𝑧 ∈ {𝑧 ∣ ¬ 𝜑})
11	5, 10	mpgbir 1796	1 ⊢ (V ∖ {𝑧 ∣ 𝜑}) = {𝑧 ∣ ¬ 𝜑}

Syntax hints:  ¬ wn 3   ↔ wb 208   = wceq 1533   ∈ wcel 2110  {cab 2799  Vcvv 3494   ∖ cdif 3932

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-dif 3938

This theorem is referenced by: (None)