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Theorem dfnelbr2 41817
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 5409 . 2 ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} ∖ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ ¬ 𝑥𝑦)}
2 df-xp 5272 . . 3 (V × V) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
3 df-eprel 5179 . . 3 E = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}
42, 3difeq12i 3869 . 2 ((V × V) ∖ E ) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} ∖ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦})
5 df-nelbr 41816 . . 3 _∉ = {⟨𝑥, 𝑦⟩ ∣ ¬ 𝑥𝑦}
6 vex 3343 . . . . . 6 𝑥 ∈ V
7 vex 3343 . . . . . 6 𝑦 ∈ V
86, 7pm3.2i 470 . . . . 5 (𝑥 ∈ V ∧ 𝑦 ∈ V)
98biantrur 528 . . . 4 𝑥𝑦 ↔ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ ¬ 𝑥𝑦))
109opabbii 4869 . . 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 383   = wceq 1632  wcel 2139  Vcvv 3340  cdif 3712  {copab 4864   E cep 5178   × cxp 5264   _∉ cnelbr 41815
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-opab 4865  df-eprel 5179  df-xp 5272  df-rel 5273  df-nelbr 41816
This theorem is referenced by: (None)
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