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Mirrors > Home > ILE Home > Th. List > ltso | GIF version |
Description: 'Less than' is a strict ordering. (Contributed by NM, 19-Jan-1997.) |
Ref | Expression |
---|---|
ltso | ⊢ < Or ℝ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltnr 7865 | . . . . 5 ⊢ (𝑥 ∈ ℝ → ¬ 𝑥 < 𝑥) | |
2 | 1 | adantl 275 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → ¬ 𝑥 < 𝑥) |
3 | lttr 7862 | . . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) → ((𝑥 < 𝑦 ∧ 𝑦 < 𝑧) → 𝑥 < 𝑧)) | |
4 | 3 | adantl 275 | . . . 4 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)) → ((𝑥 < 𝑦 ∧ 𝑦 < 𝑧) → 𝑥 < 𝑧)) |
5 | 2, 4 | ispod 4234 | . . 3 ⊢ (⊤ → < Po ℝ) |
6 | 5 | mptru 1341 | . 2 ⊢ < Po ℝ |
7 | axltwlin 7856 | . . 3 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦))) | |
8 | 7 | rgen3 2522 | . 2 ⊢ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ ∀𝑧 ∈ ℝ (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦)) |
9 | df-iso 4227 | . 2 ⊢ ( < Or ℝ ↔ ( < Po ℝ ∧ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ ∀𝑧 ∈ ℝ (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦)))) | |
10 | 6, 8, 9 | mpbir2an 927 | 1 ⊢ < Or ℝ |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∨ wo 698 ∧ w3a 963 ⊤wtru 1333 ∈ wcel 1481 ∀wral 2417 class class class wbr 3937 Po wpo 4224 Or wor 4225 ℝcr 7643 < clt 7824 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-po 4226 df-iso 4227 df-xp 4553 df-pnf 7826 df-mnf 7827 df-ltxr 7829 |
This theorem is referenced by: gtso 7867 ltnsym2 7878 suprlubex 8734 fimaxq 10605 |
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