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Theorem notab 3855
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 2904 . . 3 {𝑥 ∈ V ∣ ¬ 𝜑} = {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝜑)}
2 rabab 3195 . . 3 {𝑥 ∈ V ∣ ¬ 𝜑} = {𝑥 ∣ ¬ 𝜑}
31, 2eqtr3i 2633 . 2 {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝜑)} = {𝑥 ∣ ¬ 𝜑}
4 difab 3854 . . 3 ({𝑥𝑥 ∈ V} ∖ {𝑥𝜑}) = {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝜑)}
5 abid2 2731 . . . 4 {𝑥𝑥 ∈ V} = V
65difeq1i 3685 . . 3 ({𝑥𝑥 ∈ V} ∖ {𝑥𝜑}) = (V ∖ {𝑥𝜑})
74, 6eqtr3i 2633 . 2 {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝜑)} = (V ∖ {𝑥𝜑})
83, 7eqtr3i 2633 1 {𝑥 ∣ ¬ 𝜑} = (V ∖ {𝑥𝜑})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 382   = wceq 1474  wcel 1976  {cab 2595  {crab 2899  Vcvv 3172  cdif 3536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-rab 2904  df-v 3174  df-dif 3542
This theorem is referenced by:  dfif3  4049
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