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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 12236 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) → (𝑥 ∈ (𝐴(,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴 < 𝑥 ∧ 𝑥 ≤ 𝐵))) | |
2 | 1 | biimp3a 1432 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ∧ 𝑥 ∈ (𝐴(,]𝐵)) → (𝑥 ∈ ℝ ∧ 𝐴 < 𝑥 ∧ 𝑥 ≤ 𝐵)) |
3 | 2 | simp1d 1073 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ∧ 𝑥 ∈ (𝐴(,]𝐵)) → 𝑥 ∈ ℝ) |
4 | 3 | 3expia 1267 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) → (𝑥 ∈ (𝐴(,]𝐵) → 𝑥 ∈ ℝ)) |
5 | 4 | ssrdv 3609 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) → (𝐴(,]𝐵) ⊆ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 ∈ wcel 1990 ⊆ wss 3574 class class class wbr 4653 (class class class)co 6650 ℝcr 9935 ℝ*cxr 10073 < clt 10074 ≤ cle 10075 (,]cioc 12176 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-pre-lttri 10010 ax-pre-lttrn 10011 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-ioc 12180 |
This theorem is referenced by: iocmnfcld 22572 lhop1 23777 negpitopissre 24286 eff1o 24295 dvlog2lem 24398 iocopn 39746 limcicciooub 39869 limcresiooub 39874 fourierdlem19 40343 fourierdlem33 40357 fourierdlem37 40361 fourierdlem46 40369 fourierdlem48 40371 fourierdlem49 40372 fourierdlem51 40374 fourierdlem63 40386 fourierdlem79 40402 fourierdlem89 40412 fourierdlem90 40413 fourierdlem91 40414 fourierdlem93 40416 fouriersw 40448 |
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