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Theorem vvdifopab 34340
Description: Ordered-pair class abstraction defined by a negation. (Contributed by Peter Mazsa, 25-Jun-2019.)
Assertion
Ref Expression
vvdifopab ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) = {⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑}
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem vvdifopab
StepHypRef Expression
1 opabid 5124 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
21notbii 309 . . . 4 (¬ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ¬ 𝜑)
3 opelvvdif 34339 . . . . 5 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (⟨𝑥, 𝑦⟩ ∈ ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) ↔ ¬ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}))
43el2v 34302 . . . 4 (⟨𝑥, 𝑦⟩ ∈ ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) ↔ ¬ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
5 opabid 5124 . . . 4 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑} ↔ ¬ 𝜑)
62, 4, 53bitr4i 292 . . 3 (⟨𝑥, 𝑦⟩ ∈ ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑})
76gen2 1864 . 2 𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑})
8 relxp 5275 . . . 4 Rel (V × V)
9 reldif 5386 . . . 4 (Rel (V × V) → Rel ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}))
108, 9ax-mp 5 . . 3 Rel ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
11 relopab 5395 . . 3 Rel {⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑}
12 nfcv 2894 . . . . 5 𝑥(V × V)
13 nfopab1 4863 . . . . 5 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
1412, 13nfdif 3866 . . . 4 𝑥((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
15 nfopab1 4863 . . . 4 𝑥{⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑}
16 nfcv 2894 . . . . 5 𝑦(V × V)
17 nfopab2 4864 . . . . 5 𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
1816, 17nfdif 3866 . . . 4 𝑦((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
19 nfopab2 4864 . . . 4 𝑦{⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑}
2014, 15, 18, 19eqrelf 34336 . . 3 ((Rel ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) ∧ Rel {⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑}) → (((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) = {⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑} ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑})))
2110, 11, 20mp2an 710 . 2 (((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) = {⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑} ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑}))
227, 21mpbir 221 1 ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) = {⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wal 1622   = wceq 1624  wcel 2131  Vcvv 3332  cdif 3704  cop 4319  {copab 4856   × cxp 5256  Rel wrel 5263
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pr 5047
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-opab 4857  df-xp 5264  df-rel 5265
This theorem is referenced by:  dfssr2  34564
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