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Mirrors > Home > MPE Home > Th. List > axlttri | Structured version Visualization version GIF version |
Description: Ordering on reals satisfies strict trichotomy. Axiom 18 of 22 for real and complex numbers, derived from ZF set theory. (This restates ax-pre-lttri 10768 with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005.) |
Ref | Expression |
---|---|
axlttri | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 < 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-pre-lttri 10768 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 <ℝ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 <ℝ 𝐴))) | |
2 | ltxrlt 10868 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 𝐴 <ℝ 𝐵)) | |
3 | ltxrlt 10868 | . . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐵 < 𝐴 ↔ 𝐵 <ℝ 𝐴)) | |
4 | 3 | ancoms 462 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐵 < 𝐴 ↔ 𝐵 <ℝ 𝐴)) |
5 | 4 | orbi2d 916 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 = 𝐵 ∨ 𝐵 < 𝐴) ↔ (𝐴 = 𝐵 ∨ 𝐵 <ℝ 𝐴))) |
6 | 5 | notbid 321 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (¬ (𝐴 = 𝐵 ∨ 𝐵 < 𝐴) ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 <ℝ 𝐴))) |
7 | 1, 2, 6 | 3bitr4d 314 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 < 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 847 = wceq 1543 ∈ wcel 2112 class class class wbr 5039 ℝcr 10693 <ℝ cltrr 10698 < clt 10832 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-resscn 10751 ax-pre-lttri 10768 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-ltxr 10837 |
This theorem is referenced by: ltso 10878 leloe 10884 ltnsym 10895 ltadd2 10901 lttrid 10935 ltord1 11323 recgt0 11643 recgt0ii 11703 arch 12052 xrlttri 12694 subgmulg 18511 cosord 25374 logdivlt 25463 metakunt1 39788 digexp 45569 |
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