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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfnelbr2 | Structured version Visualization version GIF version |
Description: Alternate definition of the negated membership as binary relation. (Proposed by BJ, 27-Dec-2021.) (Contributed by AV, 27-Dec-2021.) |
Ref | Expression |
---|---|
dfnelbr2 | ⊢ _∉ = ((V × V) ∖ E ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difopab 5843 | . 2 ⊢ ({〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} ∖ {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦}) = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ ¬ 𝑥 ∈ 𝑦)} | |
2 | df-xp 5695 | . . 3 ⊢ (V × V) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} | |
3 | df-eprel 5589 | . . 3 ⊢ E = {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦} | |
4 | 2, 3 | difeq12i 4134 | . 2 ⊢ ((V × V) ∖ E ) = ({〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} ∖ {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦}) |
5 | df-nelbr 47222 | . . 3 ⊢ _∉ = {〈𝑥, 𝑦〉 ∣ ¬ 𝑥 ∈ 𝑦} | |
6 | vex 3482 | . . . . . 6 ⊢ 𝑥 ∈ V | |
7 | vex 3482 | . . . . . 6 ⊢ 𝑦 ∈ V | |
8 | 6, 7 | pm3.2i 470 | . . . . 5 ⊢ (𝑥 ∈ V ∧ 𝑦 ∈ V) |
9 | 8 | biantrur 530 | . . . 4 ⊢ (¬ 𝑥 ∈ 𝑦 ↔ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ ¬ 𝑥 ∈ 𝑦)) |
10 | 9 | opabbii 5215 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ ¬ 𝑥 ∈ 𝑦} = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ ¬ 𝑥 ∈ 𝑦)} |
11 | 5, 10 | eqtri 2763 | . 2 ⊢ _∉ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ ¬ 𝑥 ∈ 𝑦)} |
12 | 1, 4, 11 | 3eqtr4ri 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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