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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 9535 | . . 3 ⊢ (𝐾 ∈ (ℤ≥‘𝑁) → 𝑁 ∈ ℤ) | |
2 | 1 | adantl 277 | . 2 ⊢ ((𝑀 ∈ (ℤ≥‘𝐾) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → 𝑁 ∈ ℤ) |
3 | eluzelz 9539 | . . 3 ⊢ (𝑀 ∈ (ℤ≥‘𝐾) → 𝑀 ∈ ℤ) | |
4 | 3 | adantr 276 | . 2 ⊢ ((𝑀 ∈ (ℤ≥‘𝐾) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → 𝑀 ∈ ℤ) |
5 | eluzle 9542 | . . . 4 ⊢ (𝐾 ∈ (ℤ≥‘𝑁) → 𝑁 ≤ 𝐾) | |
6 | 5 | adantl 277 | . . 3 ⊢ ((𝑀 ∈ (ℤ≥‘𝐾) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → 𝑁 ≤ 𝐾) |
7 | eluzle 9542 | . . . 4 ⊢ (𝑀 ∈ (ℤ≥‘𝐾) → 𝐾 ≤ 𝑀) | |
8 | 7 | adantr 276 | . . 3 ⊢ ((𝑀 ∈ (ℤ≥‘𝐾) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → 𝐾 ≤ 𝑀) |
9 | eluzelz 9539 | . . . . 5 ⊢ (𝐾 ∈ (ℤ≥‘𝑁) → 𝐾 ∈ ℤ) | |
10 | 9 | adantl 277 | . . . 4 ⊢ ((𝑀 ∈ (ℤ≥‘𝐾) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → 𝐾 ∈ ℤ) |
11 | zletr 9304 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((𝑁 ≤ 𝐾 ∧ 𝐾 ≤ 𝑀) → 𝑁 ≤ 𝑀)) | |
12 | 2, 10, 4, 11 | syl3anc 1238 | . . 3 ⊢ ((𝑀 ∈ (ℤ≥‘𝐾) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → ((𝑁 ≤ 𝐾 ∧ 𝐾 ≤ 𝑀) → 𝑁 ≤ 𝑀)) |
13 | 6, 8, 12 | mp2and 433 | . 2 ⊢ ((𝑀 ∈ (ℤ≥‘𝐾) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → 𝑁 ≤ 𝑀) |
14 | eluz2 9536 | . 2 ⊢ (𝑀 ∈ (ℤ≥‘𝑁) ↔ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ≤ 𝑀)) | |
15 | 2, 4, 13, 14 | syl3anbrc 1181 | 1 ⊢ ((𝑀 ∈ (ℤ≥‘𝐾) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → 𝑀 ∈ (ℤ≥‘𝑁)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∈ wcel 2148 class class class wbr 4005 ‘cfv 5218 ≤ cle 7995 ℤcz 9255 ℤ≥cuz 9530 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-cnex 7904 ax-resscn 7905 ax-pre-ltwlin 7926 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2741 df-sbc 2965 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-fv 5226 df-ov 5880 df-pnf 7996 df-mnf 7997 df-xr 7998 df-ltxr 7999 df-le 8000 df-neg 8133 df-z 9256 df-uz 9531 |
This theorem is referenced by: uztrn2 9547 fzsplit2 10052 fzass4 10064 fzss1 10065 fzss2 10066 uzsplit 10094 seq3fveq2 10471 ser3mono 10480 seq3split 10481 seq3f1olemqsumkj 10500 seq3f1olemqsumk 10501 seq3id 10510 seq3id2 10511 seq3z 10513 seq3coll 10824 cvgratgt0 11543 mertenslemi1 11545 zproddc 11589 dvdsfac 11868 |
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