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Mirrors > Home > MPE Home > Th. List > xrlttri2 | Structured version Visualization version GIF version |
Description: Trichotomy law for 'less than' for extended reals. (Contributed by NM, 10-Dec-2007.) |
Ref | Expression |
---|---|
xrlttri2 | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ≠ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrltso 12886 | . . . 4 ⊢ < Or ℝ* | |
2 | sotrieq 5533 | . . . 4 ⊢ (( < Or ℝ* ∧ (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)) → (𝐴 = 𝐵 ↔ ¬ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))) | |
3 | 1, 2 | mpan 687 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 = 𝐵 ↔ ¬ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))) |
4 | 3 | bicomd 222 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (¬ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴) ↔ 𝐴 = 𝐵)) |
5 | 4 | necon1abid 2984 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ≠ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 844 = wceq 1542 ∈ wcel 2110 ≠ wne 2945 class class class wbr 5079 Or wor 5503 ℝ*cxr 11019 < clt 11020 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7583 ax-cnex 10938 ax-resscn 10939 ax-pre-lttri 10956 ax-pre-lttrn 10957 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-br 5080 df-opab 5142 df-mpt 5163 df-id 5490 df-po 5504 df-so 5505 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-er 8490 df-en 8726 df-dom 8727 df-sdom 8728 df-pnf 11022 df-mnf 11023 df-xr 11024 df-ltxr 11025 |
This theorem is referenced by: qextlt 12948 qextle 12949 nmlnogt0 29168 sgncl 32514 sgn3da 32517 |
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