Users' Mathboxes Mathbox for Andrew Salmon < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  compab Structured version   Visualization version   GIF version

Theorem compab 38471
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 2763 . . . 4 𝑧V
2 nfab1 2765 . . . 4 𝑧{𝑧𝜑}
31, 2nfdif 3729 . . 3 𝑧(V ∖ {𝑧𝜑})
4 nfab1 2765 . . 3 𝑧{𝑧 ∣ ¬ 𝜑}
53, 4cleqf 2789 . 2 ((V ∖ {𝑧𝜑}) = {𝑧 ∣ ¬ 𝜑} ↔ ∀𝑧(𝑧 ∈ (V ∖ {𝑧𝜑}) ↔ 𝑧 ∈ {𝑧 ∣ ¬ 𝜑}))
6 abid 2609 . . . 4 (𝑧 ∈ {𝑧𝜑} ↔ 𝜑)
76notbii 310 . . 3 𝑧 ∈ {𝑧𝜑} ↔ ¬ 𝜑)
8 compel 38467 . . 3 (𝑧 ∈ (V ∖ {𝑧𝜑}) ↔ ¬ 𝑧 ∈ {𝑧𝜑})
9 abid 2609 . . 3 (𝑧 ∈ {𝑧 ∣ ¬ 𝜑} ↔ ¬ 𝜑)
107, 8, 93bitr4i 292 . 2 (𝑧 ∈ (V ∖ {𝑧𝜑}) ↔ 𝑧 ∈ {𝑧 ∣ ¬ 𝜑})
115, 10mpgbir 1725 1 (V ∖ {𝑧𝜑}) = {𝑧 ∣ ¬ 𝜑}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196   = wceq 1482  wcel 1989  {cab 2607  Vcvv 3198  cdif 3569
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-rab 2920  df-v 3200  df-dif 3575
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator