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Mirrors > Home > MPE Home > Th. List > iocssre | Structured version Visualization version GIF version |
Description: A closed-above interval with real upper bound is a set of reals. (Contributed by FL, 29-May-2014.) |
Ref | Expression |
---|---|
iocssre | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) → (𝐴(,]𝐵) ⊆ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elioc2 12802 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) → (𝑥 ∈ (𝐴(,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴 < 𝑥 ∧ 𝑥 ≤ 𝐵))) | |
2 | 1 | biimp3a 1465 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ∧ 𝑥 ∈ (𝐴(,]𝐵)) → (𝑥 ∈ ℝ ∧ 𝐴 < 𝑥 ∧ 𝑥 ≤ 𝐵)) |
3 | 2 | simp1d 1138 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ∧ 𝑥 ∈ (𝐴(,]𝐵)) → 𝑥 ∈ ℝ) |
4 | 3 | 3expia 1117 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) → (𝑥 ∈ (𝐴(,]𝐵) → 𝑥 ∈ ℝ)) |
5 | 4 | ssrdv 3975 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) → (𝐴(,]𝐵) ⊆ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 ∈ wcel 2114 ⊆ wss 3938 class class class wbr 5068 (class class class)co 7158 ℝcr 10538 ℝ*cxr 10676 < clt 10677 ≤ cle 10678 (,]cioc 12742 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-pre-lttri 10613 ax-pre-lttrn 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-ioc 12746 |
This theorem is referenced by: iocmnfcld 23379 lhop1 24613 negpitopissre 25126 eff1o 25135 dvlog2lem 25237 iocopn 41803 limcicciooub 41925 limcresiooub 41930 fourierdlem19 42418 fourierdlem33 42432 fourierdlem37 42436 fourierdlem46 42444 fourierdlem48 42446 fourierdlem49 42447 fourierdlem51 42449 fourierdlem63 42461 fourierdlem79 42477 fourierdlem89 42487 fourierdlem90 42488 fourierdlem91 42489 fourierdlem93 42491 fouriersw 42523 |
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