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Theorem vvdifopab 37906
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 5529 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
21notbii 319 . . . 4 (¬ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ¬ 𝜑)
3 opelvvdif 37905 . . . . 5 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (⟨𝑥, 𝑦⟩ ∈ ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) ↔ ¬ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}))
43el2v 3469 . . . 4 (⟨𝑥, 𝑦⟩ ∈ ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) ↔ ¬ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
5 opabidw 5529 . . . 4 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑} ↔ ¬ 𝜑)
62, 4, 53bitr4i 302 . . 3 (⟨𝑥, 𝑦⟩ ∈ ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑})
76gen2 1790 . 2 𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑})
8 relxp 5699 . . . 4 Rel (V × V)
9 reldif 5820 . . . 4 (Rel (V × V) → Rel ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}))
108, 9ax-mp 5 . . 3 Rel ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
11 relopabv 5826 . . 3 Rel {⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑}
12 nfcv 2891 . . . . 5 𝑥(V × V)
13 nfopab1 5222 . . . . 5 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
1412, 13nfdif 4123 . . . 4 𝑥((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
15 nfopab1 5222 . . . 4 𝑥{⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑}
16 nfcv 2891 . . . . 5 𝑦(V × V)
17 nfopab2 5223 . . . . 5 𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
1816, 17nfdif 4123 . . . 4 𝑦((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
19 nfopab2 5223 . . . 4 𝑦{⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑}
2014, 15, 18, 19eqrelf 37901 . . 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 690 . 2 (((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) = {⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑} ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑}))
227, 21mpbir 230 1 ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) = {⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wal 1531   = wceq 1533  wcel 2098  Vcvv 3461  cdif 3943  cop 4638  {copab 5214   × cxp 5679  Rel wrel 5686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5303  ax-nul 5310  ax-pr 5432
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3949  df-un 3951  df-ss 3963  df-nul 4325  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-opab 5215  df-xp 5687  df-rel 5688
This theorem is referenced by:  dfssr2  38145
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