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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 11601). (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 12819 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ) | |
2 | sstr 3975 | . . . . 5 ⊢ ((𝑆 ⊆ (𝐴[,]𝐵) ∧ (𝐴[,]𝐵) ⊆ ℝ) → 𝑆 ⊆ ℝ) | |
3 | 2 | ancoms 461 | . . . 4 ⊢ (((𝐴[,]𝐵) ⊆ ℝ ∧ 𝑆 ⊆ (𝐴[,]𝐵)) → 𝑆 ⊆ ℝ) |
4 | 1, 3 | sylan 582 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑆 ⊆ (𝐴[,]𝐵)) → 𝑆 ⊆ ℝ) |
5 | 4 | 3adant3 1128 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑆 ⊆ (𝐴[,]𝐵) ∧ 𝐶 ∈ 𝑆) → 𝑆 ⊆ ℝ) |
6 | ne0i 4300 | . . 3 ⊢ (𝐶 ∈ 𝑆 → 𝑆 ≠ ∅) | |
7 | 6 | 3ad2ant3 1131 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑆 ⊆ (𝐴[,]𝐵) ∧ 𝐶 ∈ 𝑆) → 𝑆 ≠ ∅) |
8 | simplr 767 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑆 ⊆ (𝐴[,]𝐵)) → 𝐵 ∈ ℝ) | |
9 | ssel 3961 | . . . . . . . 8 ⊢ (𝑆 ⊆ (𝐴[,]𝐵) → (𝑦 ∈ 𝑆 → 𝑦 ∈ (𝐴[,]𝐵))) | |
10 | elicc2 12802 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑦 ∈ (𝐴[,]𝐵) ↔ (𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵))) | |
11 | 10 | biimpd 231 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑦 ∈ (𝐴[,]𝐵) → (𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵))) |
12 | 9, 11 | sylan9r 511 | . . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑆 ⊆ (𝐴[,]𝐵)) → (𝑦 ∈ 𝑆 → (𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵))) |
13 | 12 | imp 409 | . . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑆 ⊆ (𝐴[,]𝐵)) ∧ 𝑦 ∈ 𝑆) → (𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵)) |
14 | 13 | simp3d 1140 | . . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑆 ⊆ (𝐴[,]𝐵)) ∧ 𝑦 ∈ 𝑆) → 𝑦 ≤ 𝐵) |
15 | 14 | ralrimiva 3182 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑆 ⊆ (𝐴[,]𝐵)) → ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝐵) |
16 | brralrspcev 5126 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝐵) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥) | |
17 | 8, 15, 16 | syl2anc 586 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑆 ⊆ (𝐴[,]𝐵)) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥) |
18 | 17 | 3adant3 1128 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑆 ⊆ (𝐴[,]𝐵) ∧ 𝐶 ∈ 𝑆) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥) |
19 | 5, 7, 18 | 3jca 1124 | 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 398 ∧ w3a 1083 ∈ wcel 2114 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 ⊆ wss 3936 ∅c0 4291 class class class wbr 5066 (class class class)co 7156 ℝcr 10536 ≤ cle 10676 [,]cicc 12742 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-pre-lttri 10611 ax-pre-lttrn 10612 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-icc 12746 |
This theorem is referenced by: supicc 12887 hoidmv1lelem1 42922 hoidmv1lelem3 42924 hoidmvlelem1 42926 |
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