Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  vvdifopab Structured version   Visualization version   GIF version

Theorem vvdifopab 34342
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 5177 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
21notbii 311 . . . 4 (¬ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ¬ 𝜑)
3 opelvvdif 34341 . . . . 5 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (⟨𝑥, 𝑦⟩ ∈ ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) ↔ ¬ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}))
43el2v 34306 . . . 4 (⟨𝑥, 𝑦⟩ ∈ ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) ↔ ¬ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
5 opabid 5177 . . . 4 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑} ↔ ¬ 𝜑)
62, 4, 53bitr4i 294 . . 3 (⟨𝑥, 𝑦⟩ ∈ ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑})
76gen2 1878 . 2 𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑})
8 relxp 5328 . . . 4 Rel (V × V)
9 reldif 5440 . . . 4 (Rel (V × V) → Rel ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}))
108, 9ax-mp 5 . . 3 Rel ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
11 relopab 5449 . . 3 Rel {⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑}
12 nfcv 2948 . . . . 5 𝑥(V × V)
13 nfopab1 4913 . . . . 5 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
1412, 13nfdif 3930 . . . 4 𝑥((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
15 nfopab1 4913 . . . 4 𝑥{⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑}
16 nfcv 2948 . . . . 5 𝑦(V × V)
17 nfopab2 4914 . . . . 5 𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
1816, 17nfdif 3930 . . . 4 𝑦((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
19 nfopab2 4914 . . . 4 𝑦{⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑}
2014, 15, 18, 19eqrelf 34338 . . 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 675 . 2 (((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) = {⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑} ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑}))
227, 21mpbir 222 1 ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) = {⟨𝑥, 𝑦⟩ ∣ ¬ 𝜑}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 197  wal 1635   = wceq 1637  wcel 2156  Vcvv 3391  cdif 3766  cop 4376  {copab 4906   × cxp 5309  Rel wrel 5316
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pr 5096
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-sn 4371  df-pr 4373  df-op 4377  df-opab 4907  df-xp 5317  df-rel 5318
This theorem is referenced by:  dfssr2  34562
  Copyright terms: Public domain W3C validator