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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 16291 | . 2 ⊢ ∥ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑧 ∈ ℤ (𝑧 · 𝑥) = 𝑦)} | |
| 2 | 1 | relopabiv 5830 | 1 ⊢ Rel ∥ |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 Rel wrel 5690 (class class class)co 7431 · cmul 11160 ℤcz 12613 ∥ cdvds 16290 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-ss 3968 df-opab 5206 df-xp 5691 df-rel 5692 df-dvds 16291 |
| This theorem is referenced by: nznngen 44335 nzss 44336 nzin 44337 hashnzfz 44339 |
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