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Theorem vvdifopab 35697
 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 opabidw 5377 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
21notbii 323 . . . 4 (¬ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ¬ 𝜑)
3 opelvvdif 35696 . . . . 5 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (⟨𝑥, 𝑦⟩ ∈ ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) ↔ ¬ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}))
43el2v 3448 . . . 4 (⟨𝑥, 𝑦⟩ ∈ ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) ↔ ¬ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
5 opabidw 5377 . . . 4 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑} ↔ ¬ 𝜑)
62, 4, 53bitr4i 306 . . 3 (⟨𝑥, 𝑦⟩ ∈ ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑})
76gen2 1798 . 2 𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑})
8 relxp 5537 . . . 4 Rel (V × V)
9 reldif 5652 . . . 4 (Rel (V × V) → Rel ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}))
108, 9ax-mp 5 . . 3 Rel ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
11 relopab 5660 . . 3 Rel {⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑}
12 nfcv 2955 . . . . 5 𝑥(V × V)
13 nfopab1 5099 . . . . 5 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
1412, 13nfdif 4053 . . . 4 𝑥((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
15 nfopab1 5099 . . . 4 𝑥{⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑}
16 nfcv 2955 . . . . 5 𝑦(V × V)
17 nfopab2 5100 . . . . 5 𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
1816, 17nfdif 4053 . . . 4 𝑦((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
19 nfopab2 5100 . . . 4 𝑦{⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑}
2014, 15, 18, 19eqrelf 35693 . . 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 691 . 2 (((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) = {⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑} ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑}))
227, 21mpbir 234 1 ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) = {⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑}
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 209  ∀wal 1536   = wceq 1538   ∈ wcel 2111  Vcvv 3441   ∖ cdif 3878  ⟨cop 4531  {copab 5092   × cxp 5517  Rel wrel 5524 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-opab 5093  df-xp 5525  df-rel 5526 This theorem is referenced by:  dfssr2  35915
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