Theorem notab 3234

Description: A class builder defined by a negation. (Contributed by FL, 18-Sep-2010.)

Assertion

Ref	Expression
notab	⊢ {𝑥 ∣ ¬ 𝜑} = (V ∖ {𝑥 ∣ 𝜑})

Proof of Theorem notab

Step	Hyp	Ref	Expression
1		df-rab 2357	. . 3 ⊢ {𝑥 ∈ V ∣ ¬ 𝜑} = {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝜑)}
2		rabab 2620	. . 3 ⊢ {𝑥 ∈ V ∣ ¬ 𝜑} = {𝑥 ∣ ¬ 𝜑}
3	1, 2	eqtr3i 2103	. 2 ⊢ {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝜑)} = {𝑥 ∣ ¬ 𝜑}
4		difab 3233	. . 3 ⊢ ({𝑥 ∣ 𝑥 ∈ V} ∖ {𝑥 ∣ 𝜑}) = {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝜑)}
5		abid2 2199	. . . 4 ⊢ {𝑥 ∣ 𝑥 ∈ V} = V
6	5	difeq1i 3086	. . 3 ⊢ ({𝑥 ∣ 𝑥 ∈ V} ∖ {𝑥 ∣ 𝜑}) = (V ∖ {𝑥 ∣ 𝜑})
7	4, 6	eqtr3i 2103	. 2 ⊢ {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝜑)} = (V ∖ {𝑥 ∣ 𝜑})
8	3, 7	eqtr3i 2103	1 ⊢ {𝑥 ∣ ¬ 𝜑} = (V ∖ {𝑥 ∣ 𝜑})

Syntax hints:  ¬ wn 3   ∧ wa 102   = wceq 1284   ∈ wcel 1433  {cab 2067  {crab 2352  Vcvv 2601   ∖ cdif 2970

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rab 2357  df-v 2603  df-dif 2975

This theorem is referenced by:  dfif3  3364