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Mirrors > Home > ILE Home > Th. List > axltwlin | GIF version |
Description: Real number less-than is weakly linear. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-ltwlin 7456 with ordering on the extended reals. (Contributed by Jim Kingdon, 15-Jan-2020.) |
Ref | Expression |
---|---|
axltwlin | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 → (𝐴 < 𝐶 ∨ 𝐶 < 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-pre-ltwlin 7456 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 <ℝ 𝐵 → (𝐴 <ℝ 𝐶 ∨ 𝐶 <ℝ 𝐵))) | |
2 | ltxrlt 7550 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 𝐴 <ℝ 𝐵)) | |
3 | 2 | 3adant3 963 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 ↔ 𝐴 <ℝ 𝐵)) |
4 | ltxrlt 7550 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐶 ↔ 𝐴 <ℝ 𝐶)) | |
5 | 4 | 3adant2 962 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐶 ↔ 𝐴 <ℝ 𝐶)) |
6 | ltxrlt 7550 | . . . . 5 ⊢ ((𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 < 𝐵 ↔ 𝐶 <ℝ 𝐵)) | |
7 | 6 | ancoms 264 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐶 < 𝐵 ↔ 𝐶 <ℝ 𝐵)) |
8 | 7 | 3adant1 961 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐶 < 𝐵 ↔ 𝐶 <ℝ 𝐵)) |
9 | 5, 8 | orbi12d 742 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐶 ∨ 𝐶 < 𝐵) ↔ (𝐴 <ℝ 𝐶 ∨ 𝐶 <ℝ 𝐵))) |
10 | 1, 3, 9 | 3imtr4d 201 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 → (𝐴 < 𝐶 ∨ 𝐶 < 𝐵))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 ∨ wo 664 ∧ w3a 924 ∈ wcel 1438 class class class wbr 3845 ℝcr 7347 <ℝ cltrr 7352 < clt 7520 |
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-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-pow 4009 ax-pr 4036 ax-un 4260 ax-setind 4353 ax-cnex 7434 ax-resscn 7435 ax-pre-ltwlin 7456 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-fal 1295 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ne 2256 df-nel 2351 df-ral 2364 df-rex 2365 df-rab 2368 df-v 2621 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-br 3846 df-opab 3900 df-xp 4444 df-pnf 7522 df-mnf 7523 df-ltxr 7525 |
This theorem is referenced by: ltso 7561 letr 7566 lelttr 7571 ltletr 7572 gt0add 8048 reapcotr 8073 xrltso 9264 rebtwn2zlemstep 9660 expnbnd 10073 leabs 10503 ltabs 10516 abslt 10517 absle 10518 maxabslemlub 10636 |
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