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Theorem dfnelbr2 43010
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 5595 . 2 ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} ∖ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ ¬ 𝑥𝑦)}
2 df-xp 5456 . . 3 (V × V) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
3 df-eprel 5360 . . 3 E = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}
42, 3difeq12i 4024 . 2 ((V × V) ∖ E ) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} ∖ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦})
5 df-nelbr 43009 . . 3 _∉ = {⟨𝑥, 𝑦⟩ ∣ ¬ 𝑥𝑦}
6 vex 3443 . . . . . 6 𝑥 ∈ V
7 vex 3443 . . . . . 6 𝑦 ∈ V
86, 7pm3.2i 471 . . . . 5 (𝑥 ∈ V ∧ 𝑦 ∈ V)
98biantrur 531 . . . 4 𝑥𝑦 ↔ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ ¬ 𝑥𝑦))
109opabbii 5035 . . 3 {⟨𝑥, 𝑦⟩ ∣ ¬ 𝑥𝑦} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ ¬ 𝑥𝑦)}
115, 10eqtri 2821 . 2 _∉ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ ¬ 𝑥𝑦)}
121, 4, 113eqtr4ri 2832 1 _∉ = ((V × V) ∖ E )
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 396   = wceq 1525  wcel 2083  Vcvv 3440  cdif 3862  {copab 5030   E cep 5359   × cxp 5448   _∉ cnelbr 43008
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pr 5228
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-sbc 3712  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-op 4485  df-opab 5031  df-eprel 5360  df-xp 5456  df-rel 5457  df-nelbr 43009
This theorem is referenced by: (None)
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