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Mirrors > Home > MPE Home > Th. List > Mathboxes > reldvds | Structured version Visualization version GIF version |
Description: The divides relation is in fact a relation. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
Ref | Expression |
---|---|
reldvds | ⊢ Rel ∥ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dvds 16278 | . 2 ⊢ ∥ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑧 ∈ ℤ (𝑧 · 𝑥) = 𝑦)} | |
2 | 1 | relopabiv 5828 | 1 ⊢ Rel ∥ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1535 ∈ wcel 2104 ∃wrex 3066 Rel wrel 5689 (class class class)co 7426 · cmul 11152 ℤcz 12605 ∥ cdvds 16277 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-ext 2704 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1538 df-ex 1775 df-sb 2061 df-clab 2711 df-cleq 2725 df-clel 2812 df-v 3479 df-ss 3980 df-opab 5213 df-xp 5690 df-rel 5691 df-dvds 16278 |
This theorem is referenced by: nznngen 44278 nzss 44279 nzin 44280 hashnzfz 44282 |
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