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Mirrors > Home > ILE Home > Th. List > iccssre | GIF version |
Description: A closed real interval is a set of reals. (Contributed by FL, 6-Jun-2007.) (Proof shortened by Paul Chapman, 21-Jan-2008.) |
Ref | Expression |
---|---|
iccssre | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elicc2 9689 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵))) | |
2 | 1 | biimp3a 1308 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵)) |
3 | 2 | simp1d 978 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ ℝ) |
4 | 3 | 3expia 1168 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑥 ∈ (𝐴[,]𝐵) → 𝑥 ∈ ℝ)) |
5 | 4 | ssrdv 3073 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 947 ∈ wcel 1465 ⊆ wss 3041 class class class wbr 3899 (class class class)co 5742 ℝcr 7587 ≤ cle 7769 [,]cicc 9642 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 |
This theorem depends on definitions: df-bi 116 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-id 4185 df-po 4188 df-iso 4189 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-iota 5058 df-fun 5095 df-fv 5101 df-ov 5745 df-oprab 5746 df-mpo 5747 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-icc 9646 |
This theorem is referenced by: iccsupr 9717 iccshftri 9746 iccshftli 9748 iccdili 9750 icccntri 9752 unitssre 9756 cos12dec 11401 suplociccreex 12698 suplociccex 12699 dedekindicclemuub 12700 dedekindicclemlu 12704 dedekindicclemeu 12705 dedekindicclemicc 12706 dedekindicc 12707 cosz12 12788 |
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