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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 7937 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 7937 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 <ℝ 𝐵 → (𝐴 <ℝ 𝐶 ∨ 𝐶 <ℝ 𝐵))) | |
2 | ltxrlt 8036 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 𝐴 <ℝ 𝐵)) | |
3 | 2 | 3adant3 1018 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 ↔ 𝐴 <ℝ 𝐵)) |
4 | ltxrlt 8036 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐶 ↔ 𝐴 <ℝ 𝐶)) | |
5 | 4 | 3adant2 1017 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐶 ↔ 𝐴 <ℝ 𝐶)) |
6 | ltxrlt 8036 | . . . . 5 ⊢ ((𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 < 𝐵 ↔ 𝐶 <ℝ 𝐵)) | |
7 | 6 | ancoms 268 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐶 < 𝐵 ↔ 𝐶 <ℝ 𝐵)) |
8 | 7 | 3adant1 1016 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐶 < 𝐵 ↔ 𝐶 <ℝ 𝐵)) |
9 | 5, 8 | orbi12d 794 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐶 ∨ 𝐶 < 𝐵) ↔ (𝐴 <ℝ 𝐶 ∨ 𝐶 <ℝ 𝐵))) |
10 | 1, 3, 9 | 3imtr4d 203 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 → (𝐴 < 𝐶 ∨ 𝐶 < 𝐵))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 ∨ wo 709 ∧ w3a 979 ∈ wcel 2158 class class class wbr 4015 ℝcr 7823 <ℝ cltrr 7828 < clt 8005 |
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 7915 ax-resscn 7916 ax-pre-ltwlin 7937 |
This theorem depends on definitions: df-bi 117 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-pnf 8007 df-mnf 8008 df-ltxr 8010 |
This theorem is referenced by: ltso 8048 letr 8053 lelttr 8059 ltletr 8060 gt0add 8543 reapcotr 8568 sup3exmid 8927 xrltso 9809 rebtwn2zlemstep 10266 expnbnd 10657 leabs 11096 ltabs 11109 abslt 11110 absle 11111 maxabslemlub 11229 suplociccreex 14342 ivthinclemloc 14359 cnplimclemle 14377 reeff1o 14434 efltlemlt 14435 sin0pilem2 14443 coseq0negpitopi 14497 cos02pilt1 14512 |
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