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Mirrors > Home > MPE Home > Th. List > Mathboxes > xrleneltd | Structured version Visualization version GIF version |
Description: 'Less than or equal to' and 'not equals' implies 'less than', for extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
xrleneltd.a | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
xrleneltd.b | ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
xrleneltd.alb | ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
xrleneltd.anb | ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
Ref | Expression |
---|---|
xrleneltd | ⊢ (𝜑 → 𝐴 < 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrleneltd.anb | . . 3 ⊢ (𝜑 → 𝐴 ≠ 𝐵) | |
2 | 1 | necomd 3001 | . 2 ⊢ (𝜑 → 𝐵 ≠ 𝐴) |
3 | xrleneltd.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
4 | xrleneltd.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
5 | xrleneltd.alb | . . 3 ⊢ (𝜑 → 𝐴 ≤ 𝐵) | |
6 | xrleltne 12878 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → (𝐴 < 𝐵 ↔ 𝐵 ≠ 𝐴)) | |
7 | 3, 4, 5, 6 | syl3anc 1370 | . 2 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ 𝐵 ≠ 𝐴)) |
8 | 2, 7 | mpbird 256 | 1 ⊢ (𝜑 → 𝐴 < 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∈ wcel 2110 ≠ wne 2945 class class class wbr 5079 ℝ*cxr 11009 < clt 11010 ≤ cle 11011 |
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 7582 ax-cnex 10928 ax-resscn 10929 ax-pre-lttri 10946 ax-pre-lttrn 10947 |
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 8481 df-en 8717 df-dom 8718 df-sdom 8719 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 |
This theorem is referenced by: infleinf 42882 eliccelicod 43039 ge0xrre 43040 ressioosup 43064 ressiooinf 43066 sge0pr 43903 |
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