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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 11082 | . . 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 956 | 1 ⊢ (𝜑 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∨ wo 844 ∈ wcel 2106 class class class wbr 5074 ℝcr 10870 < clt 11009 ≤ cle 11010 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-br 5075 df-opab 5137 df-xp 5595 df-cnv 5597 df-xr 11013 df-le 11015 |
This theorem is referenced by: iccsplit 13217 expnbnd 13947 hashf1 14171 absmax 15041 sinltx 15898 iccntr 23984 pmltpclem2 24613 cniccbdd 24625 iccvolcl 24731 ioovolcl 24734 dyaddisjlem 24759 mbfposr 24816 itg1ge0a 24876 itg2monolem1 24915 itgioo 24980 c1lip1 25161 plyeq0lem 25371 aalioulem5 25496 pserulm 25581 tanord 25694 birthdaylem3 26103 fsumharmonic 26161 chpo1ubb 26629 mblfinlem2 35815 metakunt9 40133 ioodvbdlimc1 43474 ioodvbdlimc2 43476 ibliooicc 43512 fourierdlem107 43754 |
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