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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 5701 | . 2 ⊢ ({〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} ∖ {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦}) = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ ¬ 𝑥 ∈ 𝑦)} | |
2 | df-xp 5560 | . . 3 ⊢ (V × V) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} | |
3 | df-eprel 5464 | . . 3 ⊢ E = {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦} | |
4 | 2, 3 | difeq12i 4096 | . 2 ⊢ ((V × V) ∖ E ) = ({〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} ∖ {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦}) |
5 | df-nelbr 43470 | . . 3 ⊢ _∉ = {〈𝑥, 𝑦〉 ∣ ¬ 𝑥 ∈ 𝑦} | |
6 | vex 3497 | . . . . . 6 ⊢ 𝑥 ∈ V | |
7 | vex 3497 | . . . . . 6 ⊢ 𝑦 ∈ V | |
8 | 6, 7 | pm3.2i 473 | . . . . 5 ⊢ (𝑥 ∈ V ∧ 𝑦 ∈ V) |
9 | 8 | biantrur 533 | . . . 4 ⊢ (¬ 𝑥 ∈ 𝑦 ↔ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ ¬ 𝑥 ∈ 𝑦)) |
10 | 9 | opabbii 5132 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ ¬ 𝑥 ∈ 𝑦} = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ ¬ 𝑥 ∈ 𝑦)} |
11 | 5, 10 | eqtri 2844 | . 2 ⊢ _∉ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ ¬ 𝑥 ∈ 𝑦)} |
12 | 1, 4, 11 | 3eqtr4ri 2855 | 1 ⊢ _∉ = ((V × V) ∖ E ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ∖ cdif 3932 {copab 5127 E cep 5463 × cxp 5552 _∉ cnelbr 43469 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-opab 5128 df-eprel 5464 df-xp 5560 df-rel 5561 df-nelbr 43470 |
This theorem is referenced by: (None) |
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