![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > fzssp1 | Structured version Visualization version GIF version |
Description: Subset relationship for finite sets of sequential integers. (Contributed by NM, 21-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
fzssp1 | ⊢ (𝑀...𝑁) ⊆ (𝑀...(𝑁 + 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzel2 13558 | . . . 4 ⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑁 ∈ ℤ) | |
2 | uzid 12890 | . . . 4 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ≥‘𝑁)) | |
3 | peano2uz 12940 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑁) → (𝑁 + 1) ∈ (ℤ≥‘𝑁)) | |
4 | fzss2 13600 | . . . 4 ⊢ ((𝑁 + 1) ∈ (ℤ≥‘𝑁) → (𝑀...𝑁) ⊆ (𝑀...(𝑁 + 1))) | |
5 | 1, 2, 3, 4 | 4syl 19 | . . 3 ⊢ (𝑘 ∈ (𝑀...𝑁) → (𝑀...𝑁) ⊆ (𝑀...(𝑁 + 1))) |
6 | id 22 | . . 3 ⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑘 ∈ (𝑀...𝑁)) | |
7 | 5, 6 | sseldd 3995 | . 2 ⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑘 ∈ (𝑀...(𝑁 + 1))) |
8 | 7 | ssriv 3998 | 1 ⊢ (𝑀...𝑁) ⊆ (𝑀...(𝑁 + 1)) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 ⊆ wss 3962 ‘cfv 6562 (class class class)co 7430 1c1 11153 + caddc 11155 ℤcz 12610 ℤ≥cuz 12875 ...cfz 13543 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-n0 12524 df-z 12611 df-uz 12876 df-fz 13544 |
This theorem is referenced by: fzelp1 13612 fseq1p1m1 13634 monoord2 14070 seqf1olem1 14078 seqf1olem2 14079 seqz 14087 binomlem 15861 binom1dif 15865 bpolycl 16084 bpolysum 16085 bpolydiflem 16086 bpoly4 16091 gsumsplit1r 18712 freshmansdream 21610 1stcfb 23468 axlowdimlem13 28983 axlowdimlem16 28986 gsumnunsn 34534 pthhashvtx 35111 cvmliftlem7 35275 poimirlem3 37609 poimirlem4 37610 volsupnfl 37651 sdclem2 37728 fdc 37731 mettrifi 37743 mapfzcons1cl 42705 2rexfrabdioph 42783 3rexfrabdioph 42784 4rexfrabdioph 42785 6rexfrabdioph 42786 7rexfrabdioph 42787 rabdiophlem2 42789 jm2.27dlem5 43001 monoord2xrv 45433 stoweidlem11 45966 stoweidlem34 45989 carageniuncllem1 46476 |
Copyright terms: Public domain | W3C validator |