Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > ioogtlbd | Structured version Visualization version GIF version |
Description: An element of a closed interval is greater than its lower bound. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
ioogtlbd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
ioogtlbd.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
ioogtlbd.3 | ⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵)) |
Ref | Expression |
---|---|
ioogtlbd | ⊢ (𝜑 → 𝐴 < 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ioogtlbd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
2 | ioogtlbd.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
3 | ioogtlbd.3 | . 2 ⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵)) | |
4 | ioogtlb 42996 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴(,)𝐵)) → 𝐴 < 𝐶) | |
5 | 1, 2, 3, 4 | syl3anc 1370 | 1 ⊢ (𝜑 → 𝐴 < 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 class class class wbr 5079 (class class class)co 7269 ℝ*cxr 11001 < clt 11002 (,)cioo 13070 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7580 ax-cnex 10920 ax-resscn 10921 ax-pre-lttri 10938 ax-pre-lttrn 10939 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-id 5489 df-po 5503 df-so 5504 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-ov 7272 df-oprab 7273 df-mpo 7274 df-1st 7818 df-2nd 7819 df-er 8473 df-en 8709 df-dom 8710 df-sdom 8711 df-pnf 11004 df-mnf 11005 df-xr 11006 df-ltxr 11007 df-le 11008 df-ioo 13074 |
This theorem is referenced by: fourierdlem60 43670 fourierdlem61 43671 pimrecltneg 44220 smfmullem1 44285 |
Copyright terms: Public domain | W3C validator |