Theorem compab 44454

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 2905	. . . 4 ⊢ Ⅎ𝑧V
2		nfab1 2907	. . . 4 ⊢ Ⅎ𝑧{𝑧 ∣ 𝜑}
3	1, 2	nfdif 4142	. . 3 ⊢ Ⅎ𝑧(V ∖ {𝑧 ∣ 𝜑})
4		nfab1 2907	. . 3 ⊢ Ⅎ𝑧{𝑧 ∣ ¬ 𝜑}
5	3, 4	cleqf 2934	. 2 ⊢ ((V ∖ {𝑧 ∣ 𝜑}) = {𝑧 ∣ ¬ 𝜑} ↔ ∀𝑧(𝑧 ∈ (V ∖ {𝑧 ∣ 𝜑}) ↔ 𝑧 ∈ {𝑧 ∣ ¬ 𝜑}))
6		abid 2718	. . . 4 ⊢ (𝑧 ∈ {𝑧 ∣ 𝜑} ↔ 𝜑)
7	6	notbii 320	. . 3 ⊢ (¬ 𝑧 ∈ {𝑧 ∣ 𝜑} ↔ ¬ 𝜑)
8		velcomp 3981	. . 3 ⊢ (𝑧 ∈ (V ∖ {𝑧 ∣ 𝜑}) ↔ ¬ 𝑧 ∈ {𝑧 ∣ 𝜑})
9		abid 2718	. . 3 ⊢ (𝑧 ∈ {𝑧 ∣ ¬ 𝜑} ↔ ¬ 𝜑)
10	7, 8, 9	3bitr4i 303	. 2 ⊢ (𝑧 ∈ (V ∖ {𝑧 ∣ 𝜑}) ↔ 𝑧 ∈ {𝑧 ∣ ¬ 𝜑})
11	5, 10	mpgbir 1798	1 ⊢ (V ∖ {𝑧 ∣ 𝜑}) = {𝑧 ∣ ¬ 𝜑}

Syntax hints:  ¬ wn 3   ↔ wb 206   = wceq 1539   ∈ wcel 2108  {cab 2714  Vcvv 3481   ∖ cdif 3963

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1542  df-ex 1779  df-nf 1783  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-v 3483  df-dif 3969

This theorem is referenced by: (None)