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Mirrors > Home > MPE Home > Th. List > lecasei | Structured version Visualization version GIF version |
Description: Ordering elimination by cases. (Contributed by NM, 6-Jul-2007.) |
Ref | Expression |
---|---|
lecase.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
lecase.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
lecase.3 | ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝜓) |
lecase.4 | ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → 𝜓) |
Ref | Expression |
---|---|
lecasei | ⊢ (𝜑 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lecase.3 | . 2 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝜓) | |
2 | lecase.4 | . 2 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → 𝜓) | |
3 | lecase.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
4 | lecase.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
5 | letric 10740 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴)) | |
6 | 3, 4, 5 | syl2anc 586 | . 2 ⊢ (𝜑 → (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴)) |
7 | 1, 2, 6 | mpjaodan 955 | 1 ⊢ (𝜑 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∨ wo 843 ∈ wcel 2114 class class class wbr 5066 ℝcr 10536 ≤ cle 10676 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-pre-lttri 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 |
This theorem is referenced by: wloglei 11172 nn2ge 11665 max0sub 12590 leabs 14659 max0add 14670 limsupgre 14838 ntrivcvgmul 15258 1arithlem4 16262 mndodcong 18670 metustto 23163 reconn 23436 dyaddisj 24197 volcn 24207 ditgcl 24456 ditgswap 24457 ditgsplit 24459 dvfsumlem3 24625 ftc2ditg 24643 coseq0negpitopi 25089 asinlem3 25449 atanlogaddlem 25491 atanlogadd 25492 ppiub 25780 dchrisum0 26096 pntrmax 26140 padicabv 26206 nacsfix 39329 acongrep 39597 hbt 39750 |
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