Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > uztrn | GIF version |
Description: Transitive law for sets of upper integers. (Contributed by NM, 20-Sep-2005.) |
Ref | Expression |
---|---|
uztrn | ⊢ ((𝑀 ∈ (ℤ≥‘𝐾) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → 𝑀 ∈ (ℤ≥‘𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluzel2 9299 | . . 3 ⊢ (𝐾 ∈ (ℤ≥‘𝑁) → 𝑁 ∈ ℤ) | |
2 | 1 | adantl 275 | . 2 ⊢ ((𝑀 ∈ (ℤ≥‘𝐾) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → 𝑁 ∈ ℤ) |
3 | eluzelz 9303 | . . 3 ⊢ (𝑀 ∈ (ℤ≥‘𝐾) → 𝑀 ∈ ℤ) | |
4 | 3 | adantr 274 | . 2 ⊢ ((𝑀 ∈ (ℤ≥‘𝐾) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → 𝑀 ∈ ℤ) |
5 | eluzle 9306 | . . . 4 ⊢ (𝐾 ∈ (ℤ≥‘𝑁) → 𝑁 ≤ 𝐾) | |
6 | 5 | adantl 275 | . . 3 ⊢ ((𝑀 ∈ (ℤ≥‘𝐾) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → 𝑁 ≤ 𝐾) |
7 | eluzle 9306 | . . . 4 ⊢ (𝑀 ∈ (ℤ≥‘𝐾) → 𝐾 ≤ 𝑀) | |
8 | 7 | adantr 274 | . . 3 ⊢ ((𝑀 ∈ (ℤ≥‘𝐾) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → 𝐾 ≤ 𝑀) |
9 | eluzelz 9303 | . . . . 5 ⊢ (𝐾 ∈ (ℤ≥‘𝑁) → 𝐾 ∈ ℤ) | |
10 | 9 | adantl 275 | . . . 4 ⊢ ((𝑀 ∈ (ℤ≥‘𝐾) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → 𝐾 ∈ ℤ) |
11 | zletr 9071 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((𝑁 ≤ 𝐾 ∧ 𝐾 ≤ 𝑀) → 𝑁 ≤ 𝑀)) | |
12 | 2, 10, 4, 11 | syl3anc 1201 | . . 3 ⊢ ((𝑀 ∈ (ℤ≥‘𝐾) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → ((𝑁 ≤ 𝐾 ∧ 𝐾 ≤ 𝑀) → 𝑁 ≤ 𝑀)) |
13 | 6, 8, 12 | mp2and 429 | . 2 ⊢ ((𝑀 ∈ (ℤ≥‘𝐾) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → 𝑁 ≤ 𝑀) |
14 | eluz2 9300 | . 2 ⊢ (𝑀 ∈ (ℤ≥‘𝑁) ↔ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ≤ 𝑀)) | |
15 | 2, 4, 13, 14 | syl3anbrc 1150 | 1 ⊢ ((𝑀 ∈ (ℤ≥‘𝐾) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → 𝑀 ∈ (ℤ≥‘𝑁)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 1465 class class class wbr 3899 ‘cfv 5093 ≤ cle 7769 ℤcz 9022 ℤ≥cuz 9294 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-pre-ltwlin 7701 |
This theorem depends on definitions: df-bi 116 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-fv 5101 df-ov 5745 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-neg 7904 df-z 9023 df-uz 9295 |
This theorem is referenced by: uztrn2 9311 fzsplit2 9798 fzass4 9810 fzss1 9811 fzss2 9812 uzsplit 9840 seq3fveq2 10210 ser3mono 10219 seq3split 10220 seq3f1olemqsumkj 10239 seq3f1olemqsumk 10240 seq3id 10249 seq3id2 10250 seq3z 10252 seq3coll 10553 cvgratgt0 11270 mertenslemi1 11272 dvdsfac 11485 |
Copyright terms: Public domain | W3C validator |