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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 13431 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))) | |
| 2 | simp3 1139 | . . 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 1118 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐶 ≤ 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 ∈ wcel 2108 class class class wbr 5143 (class class class)co 7431 ℝ*cxr 11294 ≤ cle 11296 [,]cicc 13390 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-xr 11299 df-icc 13394 |
| This theorem is referenced by: supicc 13541 supiccub 13542 supicclub 13543 oprpiece1res1 24982 ivthlem1 25486 isosctrlem1 26861 ttgcontlem1 28899 broucube 37661 mblfinlem1 37664 ftc1cnnclem 37698 ftc2nc 37709 areaquad 43228 isosctrlem1ALT 44954 lefldiveq 45304 eliccelioc 45534 iccintsng 45536 eliccnelico 45542 eliccelicod 45543 inficc 45547 iccdificc 45552 iccleubd 45561 cncfiooiccre 45910 itgioocnicc 45992 itgspltprt 45994 itgiccshift 45995 fourierdlem1 46123 fourierdlem20 46142 fourierdlem24 46146 fourierdlem25 46147 fourierdlem27 46149 fourierdlem43 46165 fourierdlem44 46166 fourierdlem50 46171 fourierdlem51 46172 fourierdlem52 46173 fourierdlem64 46185 fourierdlem73 46194 fourierdlem76 46197 fourierdlem79 46200 fourierdlem81 46202 fourierdlem92 46213 fourierdlem102 46223 fourierdlem103 46224 fourierdlem104 46225 fourierdlem114 46235 rrxsnicc 46315 salgencntex 46358 sge0p1 46429 hoidmv1lelem3 46608 hoidmvlelem1 46610 hoidmvlelem4 46613 |
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