Theorem compab 44981

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 2923	. . . 4 ⊢ Ⅎ𝑧V
2		nfab1 2925	. . . 4 ⊢ Ⅎ𝑧{𝑧 ∣ 𝜑}
3	1, 2	nfdif 4083	. . 3 ⊢ Ⅎ𝑧(V ∖ {𝑧 ∣ 𝜑})
4		nfab1 2925	. . 3 ⊢ Ⅎ𝑧{𝑧 ∣ ¬ 𝜑}
5	3, 4	cleqf 2951	. 2 ⊢ ((V ∖ {𝑧 ∣ 𝜑}) = {𝑧 ∣ ¬ 𝜑} ↔ ∀𝑧(𝑧 ∈ (V ∖ {𝑧 ∣ 𝜑}) ↔ 𝑧 ∈ {𝑧 ∣ ¬ 𝜑}))
6		abid 2743	. . . 4 ⊢ (𝑧 ∈ {𝑧 ∣ 𝜑} ↔ 𝜑)
7	6	notbii 322	. . 3 ⊢ (¬ 𝑧 ∈ {𝑧 ∣ 𝜑} ↔ ¬ 𝜑)
8		velcomp 3919	. . 3 ⊢ (𝑧 ∈ (V ∖ {𝑧 ∣ 𝜑}) ↔ ¬ 𝑧 ∈ {𝑧 ∣ 𝜑})
9		abid 2743	. . 3 ⊢ (𝑧 ∈ {𝑧 ∣ ¬ 𝜑} ↔ ¬ 𝜑)
10	7, 8, 9	3bitr4i 305	. 2 ⊢ (𝑧 ∈ (V ∖ {𝑧 ∣ 𝜑}) ↔ 𝑧 ∈ {𝑧 ∣ ¬ 𝜑})
11	5, 10	mpgbir 1818	1 ⊢ (V ∖ {𝑧 ∣ 𝜑}) = {𝑧 ∣ ¬ 𝜑}

Syntax hints:  ¬ wn 3   ↔ wb 208   = wceq 1559   ∈ wcel 2141  {cab 2739  Vcvv 3453   ∖ cdif 3901

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1562  df-ex 1799  df-nf 1803  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-v 3455  df-dif 3907

This theorem is referenced by: (None)