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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ssuzfz | Structured version Visualization version GIF version |
Description: A finite subset of the upper integers is a subset of a finite set of sequential integers. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
Ref | Expression |
---|---|
ssuzfz.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
ssuzfz.2 | ⊢ (𝜑 → 𝐴 ⊆ 𝑍) |
ssuzfz.3 | ⊢ (𝜑 → 𝐴 ∈ Fin) |
Ref | Expression |
---|---|
ssuzfz | ⊢ (𝜑 → 𝐴 ⊆ (𝑀...sup(𝐴, ℝ, < ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssuzfz.2 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ 𝑍) | |
2 | 1 | sselda 3982 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ 𝑍) |
3 | ssuzfz.1 | . . . . . . 7 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
4 | 2, 3 | eleqtrdi 2843 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ (ℤ≥‘𝑀)) |
5 | eluzel2 12826 | . . . . . 6 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | |
6 | 4, 5 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑀 ∈ ℤ) |
7 | uzssz 12842 | . . . . . . . . . 10 ⊢ (ℤ≥‘𝑀) ⊆ ℤ | |
8 | 3, 7 | eqsstri 4016 | . . . . . . . . 9 ⊢ 𝑍 ⊆ ℤ |
9 | 8 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → 𝑍 ⊆ ℤ) |
10 | 1, 9 | sstrd 3992 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ ℤ) |
11 | 10 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐴 ⊆ ℤ) |
12 | ne0i 4334 | . . . . . . . 8 ⊢ (𝑘 ∈ 𝐴 → 𝐴 ≠ ∅) | |
13 | 12 | adantl 482 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐴 ≠ ∅) |
14 | ssuzfz.3 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
15 | 14 | adantr 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐴 ∈ Fin) |
16 | suprfinzcl 12675 | . . . . . . 7 ⊢ ((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin) → sup(𝐴, ℝ, < ) ∈ 𝐴) | |
17 | 11, 13, 15, 16 | syl3anc 1371 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → sup(𝐴, ℝ, < ) ∈ 𝐴) |
18 | 11, 17 | sseldd 3983 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → sup(𝐴, ℝ, < ) ∈ ℤ) |
19 | 10 | sselda 3982 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ ℤ) |
20 | eluzle 12834 | . . . . . 6 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → 𝑀 ≤ 𝑘) | |
21 | 4, 20 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑀 ≤ 𝑘) |
22 | zssre 12564 | . . . . . . . . 9 ⊢ ℤ ⊆ ℝ | |
23 | 22 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → ℤ ⊆ ℝ) |
24 | 10, 23 | sstrd 3992 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ ℝ) |
25 | 24 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐴 ⊆ ℝ) |
26 | simpr 485 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ 𝐴) | |
27 | eqidd 2733 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → sup(𝐴, ℝ, < ) = sup(𝐴, ℝ, < )) | |
28 | 25, 15, 26, 27 | supfirege 12200 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ≤ sup(𝐴, ℝ, < )) |
29 | 6, 18, 19, 21, 28 | elfzd 13491 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ (𝑀...sup(𝐴, ℝ, < ))) |
30 | 29 | ex 413 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → 𝑘 ∈ (𝑀...sup(𝐴, ℝ, < )))) |
31 | 30 | ralrimiv 3145 | . 2 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝑘 ∈ (𝑀...sup(𝐴, ℝ, < ))) |
32 | dfss3 3970 | . 2 ⊢ (𝐴 ⊆ (𝑀...sup(𝐴, ℝ, < )) ↔ ∀𝑘 ∈ 𝐴 𝑘 ∈ (𝑀...sup(𝐴, ℝ, < ))) | |
33 | 31, 32 | sylibr 233 | 1 ⊢ (𝜑 → 𝐴 ⊆ (𝑀...sup(𝐴, ℝ, < ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 ⊆ wss 3948 ∅c0 4322 class class class wbr 5148 ‘cfv 6543 (class class class)co 7408 Fincfn 8938 supcsup 9434 ℝcr 11108 < clt 11247 ≤ cle 11248 ℤcz 12557 ℤ≥cuz 12821 ...cfz 13483 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-sup 9436 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-n0 12472 df-z 12558 df-uz 12822 df-fz 13484 |
This theorem is referenced by: sge0isum 45133 |
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