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Theorem dfnelbr2 47290
Description: Alternate definition of the negated membership as binary relation. (Proposed by BJ, 27-Dec-2021.) (Contributed by AV, 27-Dec-2021.)
Assertion
Ref Expression
dfnelbr2 _∉ = ((V × V) ∖ E )

Proof of Theorem dfnelbr2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 difopab 5839 . 2 ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} ∖ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ ¬ 𝑥𝑦)}
2 df-xp 5690 . . 3 (V × V) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
3 df-eprel 5583 . . 3 E = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}
42, 3difeq12i 4123 . 2 ((V × V) ∖ E ) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} ∖ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦})
5 df-nelbr 47289 . . 3 _∉ = {⟨𝑥, 𝑦⟩ ∣ ¬ 𝑥𝑦}
6 vex 3483 . . . . . 6 𝑥 ∈ V
7 vex 3483 . . . . . 6 𝑦 ∈ V
86, 7pm3.2i 470 . . . . 5 (𝑥 ∈ V ∧ 𝑦 ∈ V)
98biantrur 530 . . . 4 𝑥𝑦 ↔ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ ¬ 𝑥𝑦))
109opabbii 5209 . . 3 {⟨𝑥, 𝑦⟩ ∣ ¬ 𝑥𝑦} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ ¬ 𝑥𝑦)}
115, 10eqtri 2764 . 2 _∉ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ ¬ 𝑥𝑦)}
121, 4, 113eqtr4ri 2775 1 _∉ = ((V × V) ∖ E )
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395   = wceq 1539  wcel 2107  Vcvv 3479  cdif 3947  {copab 5204   E cep 5582   × cxp 5682   _∉ cnelbr 47288
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-opab 5205  df-eprel 5583  df-xp 5690  df-rel 5691  df-nelbr 47289
This theorem is referenced by: (None)
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