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Mirrors > Home > ILE Home > Th. List > elicc2i | GIF version |
Description: Inference for membership in a closed interval. (Contributed by Scott Fenton, 3-Jun-2013.) |
Ref | Expression |
---|---|
elicc2i.1 | ⊢ 𝐴 ∈ ℝ |
elicc2i.2 | ⊢ 𝐵 ∈ ℝ |
Ref | Expression |
---|---|
elicc2i | ⊢ (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elicc2i.1 | . 2 ⊢ 𝐴 ∈ ℝ | |
2 | elicc2i.2 | . 2 ⊢ 𝐵 ∈ ℝ | |
3 | elicc2 9714 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))) | |
4 | 1, 2, 3 | mp2an 422 | 1 ⊢ (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∧ w3a 962 ∈ wcel 1480 class class class wbr 3924 (class class class)co 5767 ℝcr 7612 ≤ cle 7794 [,]cicc 9667 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-id 4210 df-po 4213 df-iso 4214 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-iota 5083 df-fun 5120 df-fv 5126 df-ov 5770 df-oprab 5771 df-mpo 5772 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-icc 9671 |
This theorem is referenced by: 0elunit 9762 1elunit 9763 divelunit 9778 lincmb01cmp 9779 iccf1o 9780 sinbnd2 11450 cosbnd2 11451 coseq00topi 12905 coseq0negpitopi 12906 cosordlem 12919 cosq34lt1 12920 cos02pilt1 12921 |
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