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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 5839 | . 2 ⊢ ({〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} ∖ {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦}) = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ ¬ 𝑥 ∈ 𝑦)} | |
| 2 | df-xp 5690 | . . 3 ⊢ (V × V) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} | |
| 3 | df-eprel 5583 | . . 3 ⊢ E = {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦} | |
| 4 | 2, 3 | difeq12i 4123 | . 2 ⊢ ((V × V) ∖ E ) = ({〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} ∖ {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦}) | 
| 5 | df-nelbr 47289 | . . 3 ⊢ _∉ = {〈𝑥, 𝑦〉 ∣ ¬ 𝑥 ∈ 𝑦} | |
| 6 | vex 3483 | . . . . . 6 ⊢ 𝑥 ∈ V | |
| 7 | vex 3483 | . . . . . 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 5209 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ ¬ 𝑥 ∈ 𝑦} = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ ¬ 𝑥 ∈ 𝑦)} | 
| 11 | 5, 10 | eqtri 2764 | . 2 ⊢ _∉ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ ¬ 𝑥 ∈ 𝑦)} | 
| 12 | 1, 4, 11 | 3eqtr4ri 2775 | 1 ⊢ _∉ = ((V × V) ∖ E ) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 ∧ wa 395 = wceq 1539 ∈ wcel 2107 Vcvv 3479 ∖ cdif 3947 {copab 5204 E cep 5582 × cxp 5682 _∉ cnelbr 47288 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-opab 5205 df-eprel 5583 df-xp 5690 df-rel 5691 df-nelbr 47289 | 
| This theorem is referenced by: (None) | 
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