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Mirrors > Home > ILE Home > Th. List > dvdszrcl | GIF version |
Description: Reverse closure for the divisibility relation. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
Ref | Expression |
---|---|
dvdszrcl | ⊢ (𝑋 ∥ 𝑌 → (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dvds 10591 | . . 3 ⊢ ∥ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑧 ∈ ℤ (𝑧 · 𝑥) = 𝑦)} | |
2 | opabssxp 4473 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑧 ∈ ℤ (𝑧 · 𝑥) = 𝑦)} ⊆ (ℤ × ℤ) | |
3 | 1, 2 | eqsstri 3042 | . 2 ⊢ ∥ ⊆ (ℤ × ℤ) |
4 | 3 | brel 4451 | 1 ⊢ (𝑋 ∥ 𝑌 → (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1287 ∈ wcel 1436 ∃wrex 2356 class class class wbr 3814 {copab 3867 × cxp 4402 (class class class)co 5594 · cmul 7276 ℤcz 8660 ∥ cdvds 10590 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1379 ax-7 1380 ax-gen 1381 ax-ie1 1425 ax-ie2 1426 ax-8 1438 ax-10 1439 ax-11 1440 ax-i12 1441 ax-bndl 1442 ax-4 1443 ax-14 1448 ax-17 1462 ax-i9 1466 ax-ial 1470 ax-i5r 1471 ax-ext 2067 ax-sep 3925 ax-pow 3977 ax-pr 4003 |
This theorem depends on definitions: df-bi 115 df-3an 924 df-tru 1290 df-nf 1393 df-sb 1690 df-clab 2072 df-cleq 2078 df-clel 2081 df-nfc 2214 df-ral 2360 df-rex 2361 df-v 2616 df-un 2990 df-in 2992 df-ss 2999 df-pw 3411 df-sn 3431 df-pr 3432 df-op 3434 df-br 3815 df-opab 3869 df-xp 4410 df-dvds 10591 |
This theorem is referenced by: dvdsabseq 10642 divconjdvds 10644 evenelz 10661 4dvdseven 10711 dfgcd2 10797 dvdsmulgcd 10808 isprm3 10894 dvdsnprmd 10901 |
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