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Theorem notab 4300
Description: A class abstraction 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 3428 . . 3 {𝑥 ∈ V ∣ ¬ 𝜑} = {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝜑)}
2 rabab 3498 . . 3 {𝑥 ∈ V ∣ ¬ 𝜑} = {𝑥 ∣ ¬ 𝜑}
31, 2eqtr3i 2757 . 2 {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝜑)} = {𝑥 ∣ ¬ 𝜑}
4 difab 4296 . . 3 ({𝑥𝑥 ∈ V} ∖ {𝑥𝜑}) = {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝜑)}
5 abid2 2866 . . . 4 {𝑥𝑥 ∈ V} = V
65difeq1i 4114 . . 3 ({𝑥𝑥 ∈ V} ∖ {𝑥𝜑}) = (V ∖ {𝑥𝜑})
74, 6eqtr3i 2757 . 2 {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝜑)} = (V ∖ {𝑥𝜑})
83, 7eqtr3i 2757 1 {𝑥 ∣ ¬ 𝜑} = (V ∖ {𝑥𝜑})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395   = wceq 1534  wcel 2099  {cab 2704  {crab 3427  Vcvv 3469  cdif 3941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-12 2164  ax-ext 2698
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-rab 3428  df-v 3471  df-dif 3947
This theorem is referenced by: (None)
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