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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 13416 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))) | |
2 | simp3 1135 | . . 3 ⊢ ((𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵) → 𝐶 ≤ 𝐵) | |
3 | 1, 2 | biimtrdi 252 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) → 𝐶 ≤ 𝐵)) |
4 | 3 | 3impia 1114 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐶 ≤ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 ∈ wcel 2099 class class class wbr 5145 (class class class)co 7416 ℝ*cxr 11288 ≤ cle 11290 [,]cicc 13375 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-resscn 11206 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3776 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-br 5146 df-opab 5208 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-iota 6498 df-fun 6548 df-fv 6554 df-ov 7419 df-oprab 7420 df-mpo 7421 df-xr 11293 df-icc 13379 |
This theorem is referenced by: supicc 13526 supiccub 13527 supicclub 13528 oprpiece1res1 24964 ivthlem1 25468 isosctrlem1 26843 ttgcontlem1 28815 broucube 37368 mblfinlem1 37371 ftc1cnnclem 37405 ftc2nc 37416 areaquad 42918 isosctrlem1ALT 44647 lefldiveq 44943 eliccelioc 45175 iccintsng 45177 eliccnelico 45183 eliccelicod 45184 inficc 45188 iccdificc 45193 iccleubd 45202 cncfiooiccre 45552 itgioocnicc 45634 itgspltprt 45636 itgiccshift 45637 fourierdlem1 45765 fourierdlem20 45784 fourierdlem24 45788 fourierdlem25 45789 fourierdlem27 45791 fourierdlem43 45807 fourierdlem44 45808 fourierdlem50 45813 fourierdlem51 45814 fourierdlem52 45815 fourierdlem64 45827 fourierdlem73 45836 fourierdlem76 45839 fourierdlem79 45842 fourierdlem81 45844 fourierdlem92 45855 fourierdlem102 45865 fourierdlem103 45866 fourierdlem104 45867 fourierdlem114 45877 rrxsnicc 45957 salgencntex 46000 sge0p1 46071 hoidmv1lelem3 46250 hoidmvlelem1 46252 hoidmvlelem4 46255 |
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