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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 12090 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))) | |
2 | simp3 1056 | . . 3 ⊢ ((𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵) → 𝐶 ≤ 𝐵) | |
3 | 1, 2 | syl6bi 242 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) → 𝐶 ≤ 𝐵)) |
4 | 3 | 3impia 1253 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐶 ≤ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 ∈ wcel 1977 class class class wbr 4583 (class class class)co 6549 ℝ*cxr 9952 ≤ cle 9954 [,]cicc 12049 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-xr 9957 df-icc 12053 |
This theorem is referenced by: supicc 12191 supiccub 12192 supicclub 12193 oprpiece1res1 22558 ivthlem1 23027 isosctrlem1 24348 ttgcontlem1 25565 broucube 32613 mblfinlem1 32616 ftc1cnnclem 32653 ftc2nc 32664 areaquad 36821 isosctrlem1ALT 38192 lefldiveq 38446 eliccelioc 38594 iccintsng 38596 eliccnelico 38603 eliccelicod 38604 inficc 38608 iccdificc 38613 iccleubd 38622 cncfiooiccre 38781 itgioocnicc 38869 itgspltprt 38871 itgiccshift 38872 fourierdlem1 39001 fourierdlem20 39020 fourierdlem24 39024 fourierdlem25 39025 fourierdlem27 39027 fourierdlem43 39043 fourierdlem44 39044 fourierdlem50 39049 fourierdlem51 39050 fourierdlem52 39051 fourierdlem64 39063 fourierdlem73 39072 fourierdlem76 39075 fourierdlem79 39078 fourierdlem81 39080 fourierdlem92 39091 fourierdlem102 39101 fourierdlem103 39102 fourierdlem104 39103 fourierdlem114 39113 rrxsnicc 39196 salgencntex 39237 sge0p1 39307 hoidmv1lelem3 39483 hoidmvlelem1 39485 hoidmvlelem4 39488 |
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