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Theorem dfnelbr2 44198
 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 5672 . 2 ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} ∖ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ ¬ 𝑥𝑦)}
2 df-xp 5531 . . 3 (V × V) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
3 df-eprel 5436 . . 3 E = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}
42, 3difeq12i 4027 . 2 ((V × V) ∖ E ) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} ∖ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦})
5 df-nelbr 44197 . . 3 _∉ = {⟨𝑥, 𝑦⟩ ∣ ¬ 𝑥𝑦}
6 vex 3414 . . . . . 6 𝑥 ∈ V
7 vex 3414 . . . . . 6 𝑦 ∈ V
86, 7pm3.2i 475 . . . . 5 (𝑥 ∈ V ∧ 𝑦 ∈ V)
98biantrur 535 . . . 4 𝑥𝑦 ↔ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ ¬ 𝑥𝑦))
109opabbii 5100 . . 3 {⟨𝑥, 𝑦⟩ ∣ ¬ 𝑥𝑦} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ ¬ 𝑥𝑦)}
115, 10eqtri 2782 . 2 _∉ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ ¬ 𝑥𝑦)}
121, 4, 113eqtr4ri 2793 1 _∉ = ((V × V) ∖ E )
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 400   = wceq 1539   ∈ wcel 2112  Vcvv 3410   ∖ cdif 3856  {copab 5095   E cep 5435   × cxp 5523   _∉ cnelbr 44196 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-sbc 3698  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-sn 4524  df-pr 4526  df-op 4530  df-opab 5096  df-eprel 5436  df-xp 5531  df-rel 5532  df-nelbr 44197 This theorem is referenced by: (None)
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