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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 15601 | . 2 ⊢ ∥ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑧 ∈ ℤ (𝑧 · 𝑥) = 𝑦)} | |
2 | 1 | relopabi 5687 | 1 ⊢ Rel ∥ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∃wrex 3138 Rel wrel 5553 (class class class)co 7149 · cmul 10535 ℤcz 11975 ∥ cdvds 15600 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-rab 3146 df-v 3493 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-opab 5122 df-xp 5554 df-rel 5555 df-dvds 15601 |
This theorem is referenced by: nznngen 40723 nzss 40724 nzin 40725 hashnzfz 40727 |
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