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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 13428 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))) | |
2 | simp3 1137 | . . 3 ⊢ ((𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵) → 𝐶 ≤ 𝐵) | |
3 | 1, 2 | biimtrdi 253 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) → 𝐶 ≤ 𝐵)) |
4 | 3 | 3impia 1116 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐶 ≤ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 ∈ wcel 2106 class class class wbr 5148 (class class class)co 7431 ℝ*cxr 11292 ≤ cle 11294 [,]cicc 13387 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-xr 11297 df-icc 13391 |
This theorem is referenced by: supicc 13538 supiccub 13539 supicclub 13540 oprpiece1res1 24996 ivthlem1 25500 isosctrlem1 26876 ttgcontlem1 28914 broucube 37641 mblfinlem1 37644 ftc1cnnclem 37678 ftc2nc 37689 areaquad 43205 isosctrlem1ALT 44932 lefldiveq 45243 eliccelioc 45474 iccintsng 45476 eliccnelico 45482 eliccelicod 45483 inficc 45487 iccdificc 45492 iccleubd 45501 cncfiooiccre 45851 itgioocnicc 45933 itgspltprt 45935 itgiccshift 45936 fourierdlem1 46064 fourierdlem20 46083 fourierdlem24 46087 fourierdlem25 46088 fourierdlem27 46090 fourierdlem43 46106 fourierdlem44 46107 fourierdlem50 46112 fourierdlem51 46113 fourierdlem52 46114 fourierdlem64 46126 fourierdlem73 46135 fourierdlem76 46138 fourierdlem79 46141 fourierdlem81 46143 fourierdlem92 46154 fourierdlem102 46164 fourierdlem103 46165 fourierdlem104 46166 fourierdlem114 46176 rrxsnicc 46256 salgencntex 46299 sge0p1 46370 hoidmv1lelem3 46549 hoidmvlelem1 46551 hoidmvlelem4 46554 |
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