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Theorem compab 41084
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 ∖ {𝑧𝜑}) = {𝑧 ∣ ¬ 𝜑}

Proof of Theorem compab
StepHypRef Expression
1 nfcv 2979 . . . 4 𝑧V
2 nfab1 2981 . . . 4 𝑧{𝑧𝜑}
31, 2nfdif 4077 . . 3 𝑧(V ∖ {𝑧𝜑})
4 nfab1 2981 . . 3 𝑧{𝑧 ∣ ¬ 𝜑}
53, 4cleqf 3007 . 2 ((V ∖ {𝑧𝜑}) = {𝑧 ∣ ¬ 𝜑} ↔ ∀𝑧(𝑧 ∈ (V ∖ {𝑧𝜑}) ↔ 𝑧 ∈ {𝑧 ∣ ¬ 𝜑}))
6 abid 2804 . . . 4 (𝑧 ∈ {𝑧𝜑} ↔ 𝜑)
76notbii 323 . . 3 𝑧 ∈ {𝑧𝜑} ↔ ¬ 𝜑)
8 velcomp 3923 . . 3 (𝑧 ∈ (V ∖ {𝑧𝜑}) ↔ ¬ 𝑧 ∈ {𝑧𝜑})
9 abid 2804 . . 3 (𝑧 ∈ {𝑧 ∣ ¬ 𝜑} ↔ ¬ 𝜑)
107, 8, 93bitr4i 306 . 2 (𝑧 ∈ (V ∖ {𝑧𝜑}) ↔ 𝑧 ∈ {𝑧 ∣ ¬ 𝜑})
115, 10mpgbir 1801 1 (V ∖ {𝑧𝜑}) = {𝑧 ∣ ¬ 𝜑}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 209   = wceq 1538  wcel 2114  {cab 2800  Vcvv 3469  cdif 3905
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-rab 3139  df-v 3471  df-dif 3911
This theorem is referenced by: (None)
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