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Mirrors > Home > MPE Home > Th. List > supiccub | Structured version Visualization version GIF version |
Description: The supremum of a bounded set of real numbers is an upper bound. (Contributed by Thierry Arnoux, 20-May-2019.) |
Ref | Expression |
---|---|
supicc.1 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
supicc.2 | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
supicc.3 | ⊢ (𝜑 → 𝐴 ⊆ (𝐵[,]𝐶)) |
supicc.4 | ⊢ (𝜑 → 𝐴 ≠ ∅) |
supiccub.1 | ⊢ (𝜑 → 𝐷 ∈ 𝐴) |
Ref | Expression |
---|---|
supiccub | ⊢ (𝜑 → 𝐷 ≤ sup(𝐴, ℝ, < )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | supicc.3 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ (𝐵[,]𝐶)) | |
2 | supicc.1 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
3 | supicc.2 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
4 | iccssre 13454 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵[,]𝐶) ⊆ ℝ) | |
5 | 2, 3, 4 | syl2anc 582 | . . 3 ⊢ (𝜑 → (𝐵[,]𝐶) ⊆ ℝ) |
6 | 1, 5 | sstrd 3989 | . 2 ⊢ (𝜑 → 𝐴 ⊆ ℝ) |
7 | supicc.4 | . 2 ⊢ (𝜑 → 𝐴 ≠ ∅) | |
8 | 2 | adantr 479 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) |
9 | 8 | rexrd 11305 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) |
10 | 3 | adantr 479 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ) |
11 | 10 | rexrd 11305 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ*) |
12 | 1 | sselda 3978 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ (𝐵[,]𝐶)) |
13 | iccleub 13427 | . . . . 5 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ∧ 𝑥 ∈ (𝐵[,]𝐶)) → 𝑥 ≤ 𝐶) | |
14 | 9, 11, 12, 13 | syl3anc 1368 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ≤ 𝐶) |
15 | 14 | ralrimiva 3136 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝑥 ≤ 𝐶) |
16 | brralrspcev 5205 | . . 3 ⊢ ((𝐶 ∈ ℝ ∧ ∀𝑥 ∈ 𝐴 𝑥 ≤ 𝐶) → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑥 ≤ 𝑦) | |
17 | 3, 15, 16 | syl2anc 582 | . 2 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑥 ≤ 𝑦) |
18 | supiccub.1 | . 2 ⊢ (𝜑 → 𝐷 ∈ 𝐴) | |
19 | 6, 7, 17, 18 | suprubd 12222 | 1 ⊢ (𝜑 → 𝐷 ≤ sup(𝐴, ℝ, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∈ wcel 2099 ≠ wne 2930 ∀wral 3051 ∃wrex 3060 ⊆ wss 3946 ∅c0 4322 class class class wbr 5145 (class class class)co 7416 supcsup 9476 ℝcr 11148 ℝ*cxr 11288 < clt 11289 ≤ cle 11290 [,]cicc 13375 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 ax-pre-sup 11227 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-br 5146 df-opab 5208 df-mpt 5229 df-id 5572 df-po 5586 df-so 5587 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-er 8726 df-en 8967 df-dom 8968 df-sdom 8969 df-sup 9478 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-icc 13379 |
This theorem is referenced by: (None) |
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