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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 10427 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴)) | |
6 | 3, 4, 5 | syl2anc 580 | . 2 ⊢ (𝜑 → (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴)) |
7 | 1, 2, 6 | mpjaodan 982 | 1 ⊢ (𝜑 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 ∨ wo 874 ∈ wcel 2157 class class class wbr 4843 ℝcr 10223 ≤ cle 10364 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-resscn 10281 ax-pre-lttri 10298 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-br 4844 df-opab 4906 df-mpt 4923 df-id 5220 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-er 7982 df-en 8196 df-dom 8197 df-sdom 8198 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 |
This theorem is referenced by: wloglei 10852 nn2ge 11341 max0sub 12276 leabs 14380 max0add 14391 limsupgre 14553 ntrivcvgmul 14971 1arithlem4 15963 mndodcong 18274 metustto 22686 reconn 22959 dyaddisj 23704 volcn 23714 ditgcl 23963 ditgswap 23964 ditgsplit 23966 dvfsumlem3 24132 ftc2ditg 24150 coseq0negpitopi 24597 asinlem3 24950 atanlogaddlem 24992 atanlogadd 24993 ppiub 25281 dchrisum0 25561 pntrmax 25605 padicabv 25671 nacsfix 38061 acongrep 38332 hbt 38485 |
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