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Mirrors > Home > MPE Home > Th. List > iccleub | Structured version Visualization version GIF version |
Description: An element of a closed interval is less than or equal to its upper bound. (Contributed by Jeff Hankins, 14-Jul-2009.) |
Ref | Expression |
---|---|
iccleub | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐶 ≤ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elicc1 12979 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))) | |
2 | simp3 1140 | . . 3 ⊢ ((𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵) → 𝐶 ≤ 𝐵) | |
3 | 1, 2 | syl6bi 256 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) → 𝐶 ≤ 𝐵)) |
4 | 3 | 3impia 1119 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐶 ≤ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 ∈ wcel 2110 class class class wbr 5053 (class class class)co 7213 ℝ*cxr 10866 ≤ cle 10868 [,]cicc 12938 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-sbc 3695 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-iota 6338 df-fun 6382 df-fv 6388 df-ov 7216 df-oprab 7217 df-mpo 7218 df-xr 10871 df-icc 12942 |
This theorem is referenced by: supicc 13089 supiccub 13090 supicclub 13091 oprpiece1res1 23848 ivthlem1 24348 isosctrlem1 25701 ttgcontlem1 26976 broucube 35548 mblfinlem1 35551 ftc1cnnclem 35585 ftc2nc 35596 areaquad 40750 isosctrlem1ALT 42227 lefldiveq 42504 eliccelioc 42734 iccintsng 42736 eliccnelico 42742 eliccelicod 42743 inficc 42747 iccdificc 42752 iccleubd 42761 cncfiooiccre 43111 itgioocnicc 43193 itgspltprt 43195 itgiccshift 43196 fourierdlem1 43324 fourierdlem20 43343 fourierdlem24 43347 fourierdlem25 43348 fourierdlem27 43350 fourierdlem43 43366 fourierdlem44 43367 fourierdlem50 43372 fourierdlem51 43373 fourierdlem52 43374 fourierdlem64 43386 fourierdlem73 43395 fourierdlem76 43398 fourierdlem79 43401 fourierdlem81 43403 fourierdlem92 43414 fourierdlem102 43424 fourierdlem103 43425 fourierdlem104 43426 fourierdlem114 43436 rrxsnicc 43516 salgencntex 43557 sge0p1 43627 hoidmv1lelem3 43806 hoidmvlelem1 43808 hoidmvlelem4 43811 |
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