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Theorem notab 4096
Description: A class builder defined by a negation. (Contributed by FL, 18-Sep-2010.)
Assertion
Ref Expression
notab {𝑥 ∣ ¬ 𝜑} = (V ∖ {𝑥𝜑})

Proof of Theorem notab
StepHypRef Expression
1 df-rab 3103 . . 3 {𝑥 ∈ V ∣ ¬ 𝜑} = {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝜑)}
2 rabab 3415 . . 3 {𝑥 ∈ V ∣ ¬ 𝜑} = {𝑥 ∣ ¬ 𝜑}
31, 2eqtr3i 2828 . 2 {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝜑)} = {𝑥 ∣ ¬ 𝜑}
4 difab 4095 . . 3 ({𝑥𝑥 ∈ V} ∖ {𝑥𝜑}) = {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝜑)}
5 abid2 2927 . . . 4 {𝑥𝑥 ∈ V} = V
65difeq1i 3921 . . 3 ({𝑥𝑥 ∈ V} ∖ {𝑥𝜑}) = (V ∖ {𝑥𝜑})
74, 6eqtr3i 2828 . 2 {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝜑)} = (V ∖ {𝑥𝜑})
83, 7eqtr3i 2828 1 {𝑥 ∣ ¬ 𝜑} = (V ∖ {𝑥𝜑})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 384   = wceq 1637  wcel 2156  {cab 2790  {crab 3098  Vcvv 3389  cdif 3764
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 2782
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 2791  df-cleq 2797  df-clel 2800  df-nfc 2935  df-rab 3103  df-v 3391  df-dif 3770
This theorem is referenced by:  dfif3  4291
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