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| Mirrors > Home > MPE Home > Th. List > ltlecasei | Structured version Visualization version GIF version | ||
| Description: Ordering elimination by cases. (Contributed by NM, 1-Jul-2007.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
| Ref | Expression |
|---|---|
| ltlecasei.1 | ⊢ ((𝜑 ∧ 𝐴 < 𝐵) → 𝜓) |
| ltlecasei.2 | ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → 𝜓) |
| ltlecasei.3 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
| ltlecasei.4 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
| Ref | Expression |
|---|---|
| ltlecasei | ⊢ (𝜑 → 𝜓) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ltlecasei.2 | . 2 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → 𝜓) | |
| 2 | ltlecasei.1 | . 2 ⊢ ((𝜑 ∧ 𝐴 < 𝐵) → 𝜓) | |
| 3 | ltlecasei.4 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
| 4 | ltlecasei.3 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
| 5 | lelttric 11368 | . . 3 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐵 ≤ 𝐴 ∨ 𝐴 < 𝐵)) | |
| 6 | 3, 4, 5 | syl2anc 584 | . 2 ⊢ (𝜑 → (𝐵 ≤ 𝐴 ∨ 𝐴 < 𝐵)) |
| 7 | 1, 2, 6 | mpjaodan 961 | 1 ⊢ (𝜑 → 𝜓) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∨ wo 848 ∈ wcel 2108 class class class wbr 5143 ℝcr 11154 < clt 11295 ≤ cle 11296 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-cnv 5693 df-xr 11299 df-le 11301 |
| This theorem is referenced by: iccsplit 13525 expnbnd 14271 hashf1 14496 absmax 15368 sinltx 16225 iccntr 24843 pmltpclem2 25484 cniccbdd 25496 iccvolcl 25602 ioovolcl 25605 dyaddisjlem 25630 mbfposr 25687 itg1ge0a 25746 itg2monolem1 25785 itgioo 25851 c1lip1 26036 plyeq0lem 26249 aalioulem5 26378 pserulm 26465 tanord 26580 birthdaylem3 26996 fsumharmonic 27055 chpo1ubb 27525 mblfinlem2 37665 metakunt9 42214 ioodvbdlimc1 45948 ioodvbdlimc2 45950 ibliooicc 45986 fourierdlem107 46228 |
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