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 3918 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ 𝑍) |
3 | ssuzfz.1 | . . . . . . 7 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
4 | 2, 3 | eleqtrdi 2850 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ (ℤ≥‘𝑀)) |
5 | eluzel2 12473 | . . . . . 6 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | |
6 | 4, 5 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑀 ∈ ℤ) |
7 | uzssz 12489 | . . . . . . . . . 10 ⊢ (ℤ≥‘𝑀) ⊆ ℤ | |
8 | 3, 7 | eqsstri 3952 | . . . . . . . . 9 ⊢ 𝑍 ⊆ ℤ |
9 | 8 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → 𝑍 ⊆ ℤ) |
10 | 1, 9 | sstrd 3928 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ ℤ) |
11 | 10 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐴 ⊆ ℤ) |
12 | ne0i 4266 | . . . . . . . 8 ⊢ (𝑘 ∈ 𝐴 → 𝐴 ≠ ∅) | |
13 | 12 | adantl 485 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐴 ≠ ∅) |
14 | ssuzfz.3 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
15 | 14 | adantr 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐴 ∈ Fin) |
16 | suprfinzcl 12322 | . . . . . . 7 ⊢ ((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin) → sup(𝐴, ℝ, < ) ∈ 𝐴) | |
17 | 11, 13, 15, 16 | syl3anc 1373 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → sup(𝐴, ℝ, < ) ∈ 𝐴) |
18 | 11, 17 | sseldd 3919 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → sup(𝐴, ℝ, < ) ∈ ℤ) |
19 | 10 | sselda 3918 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ ℤ) |
20 | eluzle 12481 | . . . . . 6 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → 𝑀 ≤ 𝑘) | |
21 | 4, 20 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑀 ≤ 𝑘) |
22 | zssre 12213 | . . . . . . . . 9 ⊢ ℤ ⊆ ℝ | |
23 | 22 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → ℤ ⊆ ℝ) |
24 | 10, 23 | sstrd 3928 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ ℝ) |
25 | 24 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐴 ⊆ ℝ) |
26 | simpr 488 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ 𝐴) | |
27 | eqidd 2740 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → sup(𝐴, ℝ, < ) = sup(𝐴, ℝ, < )) | |
28 | 25, 15, 26, 27 | supfirege 11849 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ≤ sup(𝐴, ℝ, < )) |
29 | 6, 18, 19, 21, 28 | elfzd 13133 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ (𝑀...sup(𝐴, ℝ, < ))) |
30 | 29 | ex 416 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → 𝑘 ∈ (𝑀...sup(𝐴, ℝ, < )))) |
31 | 30 | ralrimiv 3107 | . 2 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝑘 ∈ (𝑀...sup(𝐴, ℝ, < ))) |
32 | dfss3 3905 | . 2 ⊢ (𝐴 ⊆ (𝑀...sup(𝐴, ℝ, < )) ↔ ∀𝑘 ∈ 𝐴 𝑘 ∈ (𝑀...sup(𝐴, ℝ, < ))) | |
33 | 31, 32 | sylibr 237 | 1 ⊢ (𝜑 → 𝐴 ⊆ (𝑀...sup(𝐴, ℝ, < ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ≠ wne 2943 ∀wral 3064 ⊆ wss 3883 ∅c0 4254 class class class wbr 5070 ‘cfv 6401 (class class class)co 7235 Fincfn 8650 supcsup 9086 ℝcr 10758 < clt 10897 ≤ cle 10898 ℤcz 12206 ℤ≥cuz 12468 ...cfz 13125 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2710 ax-sep 5209 ax-nul 5216 ax-pow 5275 ax-pr 5339 ax-un 7545 ax-cnex 10815 ax-resscn 10816 ax-1cn 10817 ax-icn 10818 ax-addcl 10819 ax-addrcl 10820 ax-mulcl 10821 ax-mulrcl 10822 ax-mulcom 10823 ax-addass 10824 ax-mulass 10825 ax-distr 10826 ax-i2m1 10827 ax-1ne0 10828 ax-1rid 10829 ax-rnegex 10830 ax-rrecex 10831 ax-cnre 10832 ax-pre-lttri 10833 ax-pre-lttrn 10834 ax-pre-ltadd 10835 ax-pre-mulgt0 10836 ax-pre-sup 10837 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2818 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3425 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4255 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5153 df-tr 5179 df-id 5472 df-eprel 5478 df-po 5486 df-so 5487 df-fr 5527 df-we 5529 df-xp 5575 df-rel 5576 df-cnv 5577 df-co 5578 df-dm 5579 df-rn 5580 df-res 5581 df-ima 5582 df-pred 6179 df-ord 6237 df-on 6238 df-lim 6239 df-suc 6240 df-iota 6359 df-fun 6403 df-fn 6404 df-f 6405 df-f1 6406 df-fo 6407 df-f1o 6408 df-fv 6409 df-riota 7192 df-ov 7238 df-oprab 7239 df-mpo 7240 df-om 7667 df-1st 7783 df-2nd 7784 df-wrecs 8071 df-recs 8132 df-rdg 8170 df-er 8415 df-en 8651 df-dom 8652 df-sdom 8653 df-fin 8654 df-sup 9088 df-pnf 10899 df-mnf 10900 df-xr 10901 df-ltxr 10902 df-le 10903 df-sub 11094 df-neg 11095 df-nn 11861 df-n0 12121 df-z 12207 df-uz 12469 df-fz 13126 |
This theorem is referenced by: sge0isum 43686 |
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