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 39138
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 2948 . . . 4 𝑧V
2 nfab1 2950 . . . 4 𝑧{𝑧𝜑}
31, 2nfdif 3930 . . 3 𝑧(V ∖ {𝑧𝜑})
4 nfab1 2950 . . 3 𝑧{𝑧 ∣ ¬ 𝜑}
53, 4cleqf 2974 . 2 ((V ∖ {𝑧𝜑}) = {𝑧 ∣ ¬ 𝜑} ↔ ∀𝑧(𝑧 ∈ (V ∖ {𝑧𝜑}) ↔ 𝑧 ∈ {𝑧 ∣ ¬ 𝜑}))
6 abid 2794 . . . 4 (𝑧 ∈ {𝑧𝜑} ↔ 𝜑)
76notbii 311 . . 3 𝑧 ∈ {𝑧𝜑} ↔ ¬ 𝜑)
8 compel 39134 . . 3 (𝑧 ∈ (V ∖ {𝑧𝜑}) ↔ ¬ 𝑧 ∈ {𝑧𝜑})
9 abid 2794 . . 3 (𝑧 ∈ {𝑧 ∣ ¬ 𝜑} ↔ ¬ 𝜑)
107, 8, 93bitr4i 294 . 2 (𝑧 ∈ (V ∖ {𝑧𝜑}) ↔ 𝑧 ∈ {𝑧 ∣ ¬ 𝜑})
115, 10mpgbir 1881 1 (V ∖ {𝑧𝜑}) = {𝑧 ∣ ¬ 𝜑}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 197   = wceq 1637  wcel 2156  {cab 2792  Vcvv 3391  cdif 3766
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-rab 3105  df-v 3393  df-dif 3772
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator