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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 10735 | . . 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 952 | 1 ⊢ (𝜑 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∨ wo 841 ∈ wcel 2105 class class class wbr 5057 ℝcr 10524 < clt 10663 ≤ cle 10664 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-xp 5554 df-cnv 5556 df-xr 10667 df-le 10669 |
This theorem is referenced by: iccsplit 12859 expnbnd 13581 hashf1 13803 absmax 14677 sinltx 15530 iccntr 23356 pmltpclem2 23977 cniccbdd 23989 iccvolcl 24095 ioovolcl 24098 dyaddisjlem 24123 mbfposr 24180 itg1ge0a 24239 itg2monolem1 24278 itgioo 24343 c1lip1 24521 plyeq0lem 24727 aalioulem5 24852 pserulm 24937 tanord 25049 birthdaylem3 25458 fsumharmonic 25516 chpo1ubb 25984 mblfinlem2 34811 ioodvbdlimc1 42094 ioodvbdlimc2 42096 ibliooicc 42132 fourierdlem107 42375 |
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