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| Mirrors > Home > MPE Home > Th. List > iccsupr | Structured version Visualization version GIF version | ||
| Description: A nonempty subset of a closed real interval satisfies the conditions for the existence of its supremum (see suprcl 12228). (Contributed by Paul Chapman, 21-Jan-2008.) |
| Ref | Expression |
|---|---|
| iccsupr | ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑆 ⊆ (𝐴[,]𝐵) ∧ 𝐶 ∈ 𝑆) → (𝑆 ⊆ ℝ ∧ 𝑆 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | iccssre 13469 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ) | |
| 2 | sstr 3992 | . . . . 5 ⊢ ((𝑆 ⊆ (𝐴[,]𝐵) ∧ (𝐴[,]𝐵) ⊆ ℝ) → 𝑆 ⊆ ℝ) | |
| 3 | 2 | ancoms 458 | . . . 4 ⊢ (((𝐴[,]𝐵) ⊆ ℝ ∧ 𝑆 ⊆ (𝐴[,]𝐵)) → 𝑆 ⊆ ℝ) |
| 4 | 1, 3 | sylan 580 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑆 ⊆ (𝐴[,]𝐵)) → 𝑆 ⊆ ℝ) |
| 5 | 4 | 3adant3 1133 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑆 ⊆ (𝐴[,]𝐵) ∧ 𝐶 ∈ 𝑆) → 𝑆 ⊆ ℝ) |
| 6 | ne0i 4341 | . . 3 ⊢ (𝐶 ∈ 𝑆 → 𝑆 ≠ ∅) | |
| 7 | 6 | 3ad2ant3 1136 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑆 ⊆ (𝐴[,]𝐵) ∧ 𝐶 ∈ 𝑆) → 𝑆 ≠ ∅) |
| 8 | simplr 769 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑆 ⊆ (𝐴[,]𝐵)) → 𝐵 ∈ ℝ) | |
| 9 | ssel 3977 | . . . . . . . 8 ⊢ (𝑆 ⊆ (𝐴[,]𝐵) → (𝑦 ∈ 𝑆 → 𝑦 ∈ (𝐴[,]𝐵))) | |
| 10 | elicc2 13452 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑦 ∈ (𝐴[,]𝐵) ↔ (𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵))) | |
| 11 | 10 | biimpd 229 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑦 ∈ (𝐴[,]𝐵) → (𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵))) |
| 12 | 9, 11 | sylan9r 508 | . . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑆 ⊆ (𝐴[,]𝐵)) → (𝑦 ∈ 𝑆 → (𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵))) |
| 13 | 12 | imp 406 | . . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑆 ⊆ (𝐴[,]𝐵)) ∧ 𝑦 ∈ 𝑆) → (𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵)) |
| 14 | 13 | simp3d 1145 | . . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑆 ⊆ (𝐴[,]𝐵)) ∧ 𝑦 ∈ 𝑆) → 𝑦 ≤ 𝐵) |
| 15 | 14 | ralrimiva 3146 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑆 ⊆ (𝐴[,]𝐵)) → ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝐵) |
| 16 | brralrspcev 5203 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝐵) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥) | |
| 17 | 8, 15, 16 | syl2anc 584 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑆 ⊆ (𝐴[,]𝐵)) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥) |
| 18 | 17 | 3adant3 1133 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑆 ⊆ (𝐴[,]𝐵) ∧ 𝐶 ∈ 𝑆) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥) |
| 19 | 5, 7, 18 | 3jca 1129 | 1 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑆 ⊆ (𝐴[,]𝐵) ∧ 𝐶 ∈ 𝑆) → (𝑆 ⊆ ℝ ∧ 𝑆 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 ⊆ wss 3951 ∅c0 4333 class class class wbr 5143 (class class class)co 7431 ℝcr 11154 ≤ cle 11296 [,]cicc 13390 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-pre-lttri 11229 ax-pre-lttrn 11230 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-icc 13394 |
| This theorem is referenced by: supicc 13541 hoidmv1lelem1 46606 hoidmv1lelem3 46608 hoidmvlelem1 46610 |
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