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Mirrors > Home > MPE Home > Th. List > supfirege | Structured version Visualization version GIF version |
Description: The supremum of a finite set of real numbers is greater than or equal to all the real numbers of the set. (Contributed by AV, 1-Oct-2019.) |
Ref | Expression |
---|---|
supfirege.1 | ⊢ (𝜑 → 𝐵 ⊆ ℝ) |
supfirege.2 | ⊢ (𝜑 → 𝐵 ∈ Fin) |
supfirege.3 | ⊢ (𝜑 → 𝐶 ∈ 𝐵) |
supfirege.4 | ⊢ (𝜑 → 𝑆 = sup(𝐵, ℝ, < )) |
Ref | Expression |
---|---|
supfirege | ⊢ (𝜑 → 𝐶 ≤ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltso 11243 | . . . 4 ⊢ < Or ℝ | |
2 | 1 | a1i 11 | . . 3 ⊢ (𝜑 → < Or ℝ) |
3 | supfirege.1 | . . 3 ⊢ (𝜑 → 𝐵 ⊆ ℝ) | |
4 | supfirege.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ Fin) | |
5 | supfirege.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝐵) | |
6 | supfirege.4 | . . 3 ⊢ (𝜑 → 𝑆 = sup(𝐵, ℝ, < )) | |
7 | 2, 3, 4, 5, 6 | supgtoreq 9414 | . 2 ⊢ (𝜑 → (𝐶 < 𝑆 ∨ 𝐶 = 𝑆)) |
8 | 3, 5 | sseldd 3949 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
9 | 5 | ne0d 4299 | . . . . . 6 ⊢ (𝜑 → 𝐵 ≠ ∅) |
10 | fisupcl 9413 | . . . . . 6 ⊢ (( < Or ℝ ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵 ⊆ ℝ)) → sup(𝐵, ℝ, < ) ∈ 𝐵) | |
11 | 2, 4, 9, 3, 10 | syl13anc 1373 | . . . . 5 ⊢ (𝜑 → sup(𝐵, ℝ, < ) ∈ 𝐵) |
12 | 6, 11 | eqeltrd 2834 | . . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝐵) |
13 | 3, 12 | sseldd 3949 | . . 3 ⊢ (𝜑 → 𝑆 ∈ ℝ) |
14 | 8, 13 | leloed 11306 | . 2 ⊢ (𝜑 → (𝐶 ≤ 𝑆 ↔ (𝐶 < 𝑆 ∨ 𝐶 = 𝑆))) |
15 | 7, 14 | mpbird 257 | 1 ⊢ (𝜑 → 𝐶 ≤ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 846 = wceq 1542 ∈ wcel 2107 ≠ wne 2940 ⊆ wss 3914 ∅c0 4286 class class class wbr 5109 Or wor 5548 Fincfn 8889 supcsup 9384 ℝcr 11058 < clt 11197 ≤ cle 11198 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-resscn 11116 ax-pre-lttri 11133 ax-pre-lttrn 11134 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-om 7807 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-sup 9386 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 |
This theorem is referenced by: fsuppmapnn0fiub 13905 ssuzfz 43674 |
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