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Theorem dfnelbr2 47223
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 5843 . 2 ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} ∖ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ ¬ 𝑥𝑦)}
2 df-xp 5695 . . 3 (V × V) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
3 df-eprel 5589 . . 3 E = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}
42, 3difeq12i 4134 . 2 ((V × V) ∖ E ) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} ∖ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦})
5 df-nelbr 47222 . . 3 _∉ = {⟨𝑥, 𝑦⟩ ∣ ¬ 𝑥𝑦}
6 vex 3482 . . . . . 6 𝑥 ∈ V
7 vex 3482 . . . . . 6 𝑦 ∈ V
86, 7pm3.2i 470 . . . . 5 (𝑥 ∈ V ∧ 𝑦 ∈ V)
98biantrur 530 . . . 4 𝑥𝑦 ↔ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ ¬ 𝑥𝑦))
109opabbii 5215 . . 3 {⟨𝑥, 𝑦⟩ ∣ ¬ 𝑥𝑦} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ ¬ 𝑥𝑦)}
115, 10eqtri 2763 . 2 _∉ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ ¬ 𝑥𝑦)}
121, 4, 113eqtr4ri 2774 1 _∉ = ((V × V) ∖ E )
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  cdif 3960  {copab 5210   E cep 5588   × cxp 5687   _∉ cnelbr 47221
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-opab 5211  df-eprel 5589  df-xp 5695  df-rel 5696  df-nelbr 47222
This theorem is referenced by: (None)
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