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Mirrors > Home > MPE Home > Th. List > xrltne | Structured version Visualization version GIF version |
Description: 'Less than' implies not equal for extended reals. (Contributed by NM, 20-Jan-2006.) |
Ref | Expression |
---|---|
xrltne | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) → 𝐵 ≠ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | orc 863 | . . . 4 ⊢ (𝐴 < 𝐵 → (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)) | |
2 | xrltso 12526 | . . . . . 6 ⊢ < Or ℝ* | |
3 | sotrieq 5495 | . . . . . 6 ⊢ (( < Or ℝ* ∧ (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)) → (𝐴 = 𝐵 ↔ ¬ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))) | |
4 | 2, 3 | mpan 688 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 = 𝐵 ↔ ¬ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))) |
5 | 4 | necon2abid 3056 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴 < 𝐵 ∨ 𝐵 < 𝐴) ↔ 𝐴 ≠ 𝐵)) |
6 | 1, 5 | syl5ib 246 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 → 𝐴 ≠ 𝐵)) |
7 | 6 | 3impia 1112 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) → 𝐴 ≠ 𝐵) |
8 | 7 | necomd 3069 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) → 𝐵 ≠ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 ∧ w3a 1082 = wceq 1531 ∈ wcel 2108 ≠ wne 3014 class class class wbr 5057 Or wor 5466 ℝ*cxr 10666 < clt 10667 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-pre-lttri 10603 ax-pre-lttrn 10604 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-er 8281 df-en 8502 df-dom 8503 df-sdom 8504 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 |
This theorem is referenced by: xmulpnf1 12659 supxrbnd 12713 sgnp 14441 sgnn 14445 xrsdsreclblem 20583 supxrnemnf 30485 lfuhgr2 32358 acycgr2v 32390 xrgtned 41580 icoiccdif 41790 cncfiooicclem1 42166 |
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