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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 3074 | . 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 12541 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → (𝐴 < 𝐵 ↔ 𝐵 ≠ 𝐴)) | |
7 | 3, 4, 5, 6 | syl3anc 1367 | . 2 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ 𝐵 ≠ 𝐴)) |
8 | 2, 7 | mpbird 259 | 1 ⊢ (𝜑 → 𝐴 < 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∈ wcel 2113 ≠ wne 3019 class class class wbr 5069 ℝ*cxr 10677 < clt 10678 ≤ cle 10679 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-pre-lttri 10614 ax-pre-lttrn 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-po 5477 df-so 5478 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 |
This theorem is referenced by: infleinf 41646 eliccelicod 41812 ge0xrre 41813 ressioosup 41837 ressiooinf 41839 sge0pr 42683 |
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