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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 10016 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴)) | |
6 | 3, 4, 5 | syl2anc 691 | . 2 ⊢ (𝜑 → (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴)) |
7 | 1, 2, 6 | mpjaodan 823 | 1 ⊢ (𝜑 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 382 ∧ wa 383 ∈ wcel 1977 class class class wbr 4583 ℝcr 9814 ≤ cle 9954 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-pre-lttri 9889 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 |
This theorem is referenced by: wloglei 10439 nn2ge 10922 max0sub 11901 leabs 13887 max0add 13898 limsupgre 14060 ntrivcvgmul 14473 1arithlem4 15468 mndodcong 17784 metustto 22168 reconn 22439 dyaddisj 23170 volcn 23180 ditgcl 23428 ditgswap 23429 ditgsplit 23431 dvfsumlem3 23595 ftc2ditg 23613 coseq0negpitopi 24059 asinlem3 24398 atanlogaddlem 24440 atanlogadd 24441 ppiub 24729 dchrisum0 25009 pntrmax 25053 padicabv 25119 nacsfix 36293 acongrep 36565 hbt 36719 |
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