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Theorem notab 4333
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 3444 . . 3 {𝑥 ∈ V ∣ ¬ 𝜑} = {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝜑)}
2 rabab 3520 . . 3 {𝑥 ∈ V ∣ ¬ 𝜑} = {𝑥 ∣ ¬ 𝜑}
31, 2eqtr3i 2770 . 2 {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝜑)} = {𝑥 ∣ ¬ 𝜑}
4 difab 4329 . . 3 ({𝑥𝑥 ∈ V} ∖ {𝑥𝜑}) = {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝜑)}
5 abid2 2882 . . . 4 {𝑥𝑥 ∈ V} = V
65difeq1i 4145 . . 3 ({𝑥𝑥 ∈ V} ∖ {𝑥𝜑}) = (V ∖ {𝑥𝜑})
74, 6eqtr3i 2770 . 2 {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝜑)} = (V ∖ {𝑥𝜑})
83, 7eqtr3i 2770 1 {𝑥 ∣ ¬ 𝜑} = (V ∖ {𝑥𝜑})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395   = wceq 1537  wcel 2108  {cab 2717  {crab 3443  Vcvv 3488  cdif 3973
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 2110  ax-9 2118  ax-10 2141  ax-12 2178  ax-ext 2711
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979
This theorem is referenced by: (None)
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