![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 11365 | . . 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 960 | 1 ⊢ (𝜑 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 ∈ wcel 2105 class class class wbr 5147 ℝcr 11151 < clt 11292 ≤ cle 11293 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5148 df-opab 5210 df-xp 5694 df-cnv 5696 df-xr 11296 df-le 11298 |
This theorem is referenced by: iccsplit 13521 expnbnd 14267 hashf1 14492 absmax 15364 sinltx 16221 iccntr 24856 pmltpclem2 25497 cniccbdd 25509 iccvolcl 25615 ioovolcl 25618 dyaddisjlem 25643 mbfposr 25700 itg1ge0a 25760 itg2monolem1 25799 itgioo 25865 c1lip1 26050 plyeq0lem 26263 aalioulem5 26392 pserulm 26479 tanord 26594 birthdaylem3 27010 fsumharmonic 27069 chpo1ubb 27539 mblfinlem2 37644 metakunt9 42194 ioodvbdlimc1 45888 ioodvbdlimc2 45890 ibliooicc 45926 fourierdlem107 46168 |
Copyright terms: Public domain | W3C validator |