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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 12807 | . . 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 12874 | . . 3 ⊢ (𝜑 → sup(𝐴, ℝ, < ) ∈ (𝐵[,]𝐶)) |
7 | 1, 6 | sseldi 3962 | . 2 ⊢ (𝜑 → sup(𝐴, ℝ, < ) ∈ ℝ*) |
8 | 4, 1 | sstrdi 3976 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ ℝ*) |
9 | supiccub.1 | . . 3 ⊢ (𝜑 → 𝐷 ∈ 𝐴) | |
10 | 8, 9 | sseldd 3965 | . 2 ⊢ (𝜑 → 𝐷 ∈ ℝ*) |
11 | supicclub2.1 | . . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑧 ≤ 𝐷) | |
12 | 8 | sselda 3964 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ ℝ*) |
13 | 10 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝐷 ∈ ℝ*) |
14 | 12, 13 | xrlenltd 10695 | . . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝑧 ≤ 𝐷 ↔ ¬ 𝐷 < 𝑧)) |
15 | 11, 14 | mpbid 233 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ¬ 𝐷 < 𝑧) |
16 | 15 | nrexdv 3267 | . . 3 ⊢ (𝜑 → ¬ ∃𝑧 ∈ 𝐴 𝐷 < 𝑧) |
17 | 2, 3, 4, 5, 9 | supicclub 12876 | . . 3 ⊢ (𝜑 → (𝐷 < sup(𝐴, ℝ, < ) ↔ ∃𝑧 ∈ 𝐴 𝐷 < 𝑧)) |
18 | 16, 17 | mtbird 326 | . 2 ⊢ (𝜑 → ¬ 𝐷 < sup(𝐴, ℝ, < )) |
19 | 7, 10, 18 | xrnltled 10697 | 1 ⊢ (𝜑 → sup(𝐴, ℝ, < ) ≤ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∈ wcel 2105 ≠ wne 3013 ∃wrex 3136 ⊆ wss 3933 ∅c0 4288 class class class wbr 5057 (class class class)co 7145 supcsup 8892 ℝcr 10524 ℝ*cxr 10662 < clt 10663 ≤ cle 10664 [,]cicc 12729 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7678 df-2nd 7679 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-sup 8894 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-icc 12733 |
This theorem is referenced by: (None) |
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