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 44351
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 2904 . . . 4 𝑧V
2 nfab1 2906 . . . 4 𝑧{𝑧𝜑}
31, 2nfdif 4146 . . 3 𝑧(V ∖ {𝑧𝜑})
4 nfab1 2906 . . 3 𝑧{𝑧 ∣ ¬ 𝜑}
53, 4cleqf 2936 . 2 ((V ∖ {𝑧𝜑}) = {𝑧 ∣ ¬ 𝜑} ↔ ∀𝑧(𝑧 ∈ (V ∖ {𝑧𝜑}) ↔ 𝑧 ∈ {𝑧 ∣ ¬ 𝜑}))
6 abid 2715 . . . 4 (𝑧 ∈ {𝑧𝜑} ↔ 𝜑)
76notbii 320 . . 3 𝑧 ∈ {𝑧𝜑} ↔ ¬ 𝜑)
8 velcomp 3985 . . 3 (𝑧 ∈ (V ∖ {𝑧𝜑}) ↔ ¬ 𝑧 ∈ {𝑧𝜑})
9 abid 2715 . . 3 (𝑧 ∈ {𝑧 ∣ ¬ 𝜑} ↔ ¬ 𝜑)
107, 8, 93bitr4i 303 . 2 (𝑧 ∈ (V ∖ {𝑧𝜑}) ↔ 𝑧 ∈ {𝑧 ∣ ¬ 𝜑})
115, 10mpgbir 1797 1 (V ∖ {𝑧𝜑}) = {𝑧 ∣ ¬ 𝜑}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206   = wceq 1537  wcel 2103  {cab 2711  Vcvv 3482  cdif 3967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-v 3484  df-dif 3973
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator