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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 5595 | . 2 ⊢ ({〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} ∖ {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦}) = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ ¬ 𝑥 ∈ 𝑦)} | |
2 | df-xp 5456 | . . 3 ⊢ (V × V) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} | |
3 | df-eprel 5360 | . . 3 ⊢ E = {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦} | |
4 | 2, 3 | difeq12i 4024 | . 2 ⊢ ((V × V) ∖ E ) = ({〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} ∖ {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦}) |
5 | df-nelbr 43009 | . . 3 ⊢ _∉ = {〈𝑥, 𝑦〉 ∣ ¬ 𝑥 ∈ 𝑦} | |
6 | vex 3443 | . . . . . 6 ⊢ 𝑥 ∈ V | |
7 | vex 3443 | . . . . . 6 ⊢ 𝑦 ∈ V | |
8 | 6, 7 | pm3.2i 471 | . . . . 5 ⊢ (𝑥 ∈ V ∧ 𝑦 ∈ V) |
9 | 8 | biantrur 531 | . . . 4 ⊢ (¬ 𝑥 ∈ 𝑦 ↔ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ ¬ 𝑥 ∈ 𝑦)) |
10 | 9 | opabbii 5035 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ ¬ 𝑥 ∈ 𝑦} = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ ¬ 𝑥 ∈ 𝑦)} |
11 | 5, 10 | eqtri 2821 | . 2 ⊢ _∉ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ ¬ 𝑥 ∈ 𝑦)} |
12 | 1, 4, 11 | 3eqtr4ri 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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