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Mirrors > Home > MPE Home > Th. List > supicclub2 | Structured version Visualization version GIF version |
Description: The supremum of a bounded set of real numbers is the least upper bound. (Contributed by Thierry Arnoux, 23-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) |
supicclub2.1 | ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑧 ≤ 𝐷) |
Ref | Expression |
---|---|
supicclub2 | ⊢ (𝜑 → sup(𝐴, ℝ, < ) ≤ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iccssxr 12813 | . . 3 ⊢ (𝐵[,]𝐶) ⊆ ℝ* | |
2 | supicc.1 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
3 | supicc.2 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
4 | supicc.3 | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ (𝐵[,]𝐶)) | |
5 | supicc.4 | . . . 4 ⊢ (𝜑 → 𝐴 ≠ ∅) | |
6 | 2, 3, 4, 5 | supicc 12880 | . . 3 ⊢ (𝜑 → sup(𝐴, ℝ, < ) ∈ (𝐵[,]𝐶)) |
7 | 1, 6 | sseldi 3964 | . 2 ⊢ (𝜑 → sup(𝐴, ℝ, < ) ∈ ℝ*) |
8 | 4, 1 | sstrdi 3978 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ ℝ*) |
9 | supiccub.1 | . . 3 ⊢ (𝜑 → 𝐷 ∈ 𝐴) | |
10 | 8, 9 | sseldd 3967 | . 2 ⊢ (𝜑 → 𝐷 ∈ ℝ*) |
11 | supicclub2.1 | . . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑧 ≤ 𝐷) | |
12 | 8 | sselda 3966 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ ℝ*) |
13 | 10 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝐷 ∈ ℝ*) |
14 | 12, 13 | xrlenltd 10701 | . . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝑧 ≤ 𝐷 ↔ ¬ 𝐷 < 𝑧)) |
15 | 11, 14 | mpbid 234 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ¬ 𝐷 < 𝑧) |
16 | 15 | nrexdv 3270 | . . 3 ⊢ (𝜑 → ¬ ∃𝑧 ∈ 𝐴 𝐷 < 𝑧) |
17 | 2, 3, 4, 5, 9 | supicclub 12882 | . . 3 ⊢ (𝜑 → (𝐷 < sup(𝐴, ℝ, < ) ↔ ∃𝑧 ∈ 𝐴 𝐷 < 𝑧)) |
18 | 16, 17 | mtbird 327 | . 2 ⊢ (𝜑 → ¬ 𝐷 < sup(𝐴, ℝ, < )) |
19 | 7, 10, 18 | xrnltled 10703 | 1 ⊢ (𝜑 → sup(𝐴, ℝ, < ) ≤ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∈ wcel 2110 ≠ wne 3016 ∃wrex 3139 ⊆ wss 3935 ∅c0 4290 class class class wbr 5058 (class class class)co 7150 supcsup 8898 ℝcr 10530 ℝ*cxr 10668 < clt 10669 ≤ cle 10670 [,]cicc 12735 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-po 5468 df-so 5469 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-sup 8900 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-icc 12739 |
This theorem is referenced by: (None) |
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