![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > zletr | GIF version |
Description: Transitive law of ordering for integers. (Contributed by Alexander van der Vekens, 3-Apr-2018.) |
Ref | Expression |
---|---|
zletr | ⊢ ((𝐽 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝐿 ∈ ℤ) → ((𝐽 ≤ 𝐾 ∧ 𝐾 ≤ 𝐿) → 𝐽 ≤ 𝐿)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zre 9271 | . 2 ⊢ (𝐽 ∈ ℤ → 𝐽 ∈ ℝ) | |
2 | zre 9271 | . 2 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ ℝ) | |
3 | zre 9271 | . 2 ⊢ (𝐿 ∈ ℤ → 𝐿 ∈ ℝ) | |
4 | letr 8054 | . 2 ⊢ ((𝐽 ∈ ℝ ∧ 𝐾 ∈ ℝ ∧ 𝐿 ∈ ℝ) → ((𝐽 ≤ 𝐾 ∧ 𝐾 ≤ 𝐿) → 𝐽 ≤ 𝐿)) | |
5 | 1, 2, 3, 4 | syl3an 1290 | 1 ⊢ ((𝐽 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝐿 ∈ ℤ) → ((𝐽 ≤ 𝐾 ∧ 𝐾 ≤ 𝐿) → 𝐽 ≤ 𝐿)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 979 ∈ wcel 2158 class class class wbr 4015 ℝcr 7824 ≤ cle 8007 ℤcz 9267 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-cnex 7916 ax-resscn 7917 ax-pre-ltwlin 7938 |
This theorem depends on definitions: df-bi 117 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-rab 2474 df-v 2751 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-br 4016 df-opab 4077 df-xp 4644 df-cnv 4646 df-iota 5190 df-fv 5236 df-ov 5891 df-pnf 8008 df-mnf 8009 df-xr 8010 df-ltxr 8011 df-le 8012 df-neg 8145 df-z 9268 |
This theorem is referenced by: uztrn 9558 uzss 9562 elfz0ubfz0 10139 |
Copyright terms: Public domain | W3C validator |