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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 3948 | . . . . . . 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 12776 | . . . . . 6 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | |
6 | 4, 5 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑀 ∈ ℤ) |
7 | uzssz 12792 | . . . . . . . . . 10 ⊢ (ℤ≥‘𝑀) ⊆ ℤ | |
8 | 3, 7 | eqsstri 3982 | . . . . . . . . 9 ⊢ 𝑍 ⊆ ℤ |
9 | 8 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → 𝑍 ⊆ ℤ) |
10 | 1, 9 | sstrd 3958 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ ℤ) |
11 | 10 | adantr 482 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐴 ⊆ ℤ) |
12 | ne0i 4298 | . . . . . . . 8 ⊢ (𝑘 ∈ 𝐴 → 𝐴 ≠ ∅) | |
13 | 12 | adantl 483 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐴 ≠ ∅) |
14 | ssuzfz.3 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
15 | 14 | adantr 482 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐴 ∈ Fin) |
16 | suprfinzcl 12625 | . . . . . . 7 ⊢ ((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin) → sup(𝐴, ℝ, < ) ∈ 𝐴) | |
17 | 11, 13, 15, 16 | syl3anc 1372 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → sup(𝐴, ℝ, < ) ∈ 𝐴) |
18 | 11, 17 | sseldd 3949 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → sup(𝐴, ℝ, < ) ∈ ℤ) |
19 | 10 | sselda 3948 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ ℤ) |
20 | eluzle 12784 | . . . . . 6 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → 𝑀 ≤ 𝑘) | |
21 | 4, 20 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑀 ≤ 𝑘) |
22 | zssre 12514 | . . . . . . . . 9 ⊢ ℤ ⊆ ℝ | |
23 | 22 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → ℤ ⊆ ℝ) |
24 | 10, 23 | sstrd 3958 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ ℝ) |
25 | 24 | adantr 482 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐴 ⊆ ℝ) |
26 | simpr 486 | . . . . . 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 12150 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ≤ sup(𝐴, ℝ, < )) |
29 | 6, 18, 19, 21, 28 | elfzd 13441 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ (𝑀...sup(𝐴, ℝ, < ))) |
30 | 29 | ex 414 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → 𝑘 ∈ (𝑀...sup(𝐴, ℝ, < )))) |
31 | 30 | ralrimiv 3139 | . 2 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝑘 ∈ (𝑀...sup(𝐴, ℝ, < ))) |
32 | dfss3 3936 | . 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 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2940 ∀wral 3061 ⊆ wss 3914 ∅c0 4286 class class class wbr 5109 ‘cfv 6500 (class class class)co 7361 Fincfn 8889 supcsup 9384 ℝcr 11058 < clt 11197 ≤ cle 11198 ℤcz 12507 ℤ≥cuz 12771 ...cfz 13433 |
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 2703 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-pre-sup 11137 |
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 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 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-iun 4960 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-pred 6257 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-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 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 df-sub 11395 df-neg 11396 df-nn 12162 df-n0 12422 df-z 12508 df-uz 12772 df-fz 13434 |
This theorem is referenced by: sge0isum 44758 |
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