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Theorem compab 38962
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 2793 . . . 4 𝑧V
2 nfab1 2795 . . . 4 𝑧{𝑧𝜑}
31, 2nfdif 3764 . . 3 𝑧(V ∖ {𝑧𝜑})
4 nfab1 2795 . . 3 𝑧{𝑧 ∣ ¬ 𝜑}
53, 4cleqf 2819 . 2 ((V ∖ {𝑧𝜑}) = {𝑧 ∣ ¬ 𝜑} ↔ ∀𝑧(𝑧 ∈ (V ∖ {𝑧𝜑}) ↔ 𝑧 ∈ {𝑧 ∣ ¬ 𝜑}))
6 abid 2639 . . . 4 (𝑧 ∈ {𝑧𝜑} ↔ 𝜑)
76notbii 309 . . 3 𝑧 ∈ {𝑧𝜑} ↔ ¬ 𝜑)
8 compel 38958 . . 3 (𝑧 ∈ (V ∖ {𝑧𝜑}) ↔ ¬ 𝑧 ∈ {𝑧𝜑})
9 abid 2639 . . 3 (𝑧 ∈ {𝑧 ∣ ¬ 𝜑} ↔ ¬ 𝜑)
107, 8, 93bitr4i 292 . 2 (𝑧 ∈ (V ∖ {𝑧𝜑}) ↔ 𝑧 ∈ {𝑧 ∣ ¬ 𝜑})
115, 10mpgbir 1766 1 (V ∖ {𝑧𝜑}) = {𝑧 ∣ ¬ 𝜑}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196   = wceq 1523  wcel 2030  {cab 2637  Vcvv 3231  cdif 3604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950  df-v 3233  df-dif 3610
This theorem is referenced by: (None)
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