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Mirrors > Home > ILE Home > Th. List > frecfzen2 | Unicode version |
Description: The cardinality of a finite set of sequential integers with arbitrary endpoints. (Contributed by Jim Kingdon, 18-May-2020.) |
Ref | Expression |
---|---|
frecfzennn.1 |
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Ref | Expression |
---|---|
frecfzen2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluzel2 8254 |
. . . 4
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2 | eluzelz 8258 |
. . . 4
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3 | 1z 8047 |
. . . . 5
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4 | zsubcl 8062 |
. . . . 5
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5 | 3, 1, 4 | sylancr 393 |
. . . 4
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6 | fzen 8677 |
. . . 4
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7 | 1, 2, 5, 6 | syl3anc 1134 |
. . 3
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8 | 1 | zcnd 8137 |
. . . . 5
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9 | ax-1cn 6776 |
. . . . 5
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10 | pncan3 7016 |
. . . . 5
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11 | 8, 9, 10 | sylancl 392 |
. . . 4
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12 | zcn 8026 |
. . . . . . 7
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13 | zcn 8026 |
. . . . . . 7
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14 | addsubass 7018 |
. . . . . . . 8
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15 | 9, 14 | mp3an2 1219 |
. . . . . . 7
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16 | 12, 13, 15 | syl2an 273 |
. . . . . 6
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17 | 2, 1, 16 | syl2anc 391 |
. . . . 5
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18 | 17 | eqcomd 2042 |
. . . 4
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19 | 11, 18 | oveq12d 5473 |
. . 3
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20 | 7, 19 | breqtrd 3779 |
. 2
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21 | peano2uz 8302 |
. . 3
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22 | uznn0sub 8280 |
. . 3
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23 | frecfzennn.1 |
. . . 4
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24 | 23 | frecfzennn 8884 |
. . 3
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25 | 21, 22, 24 | 3syl 17 |
. 2
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26 | entr 6200 |
. 2
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27 | 20, 25, 26 | syl2anc 391 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-coll 3863 ax-sep 3866 ax-nul 3874 ax-pow 3918 ax-pr 3935 ax-un 4136 ax-setind 4220 ax-iinf 4254 ax-cnex 6774 ax-resscn 6775 ax-1cn 6776 ax-1re 6777 ax-icn 6778 ax-addcl 6779 ax-addrcl 6780 ax-mulcl 6781 ax-addcom 6783 ax-addass 6785 ax-distr 6787 ax-i2m1 6788 ax-0id 6791 ax-rnegex 6792 ax-cnre 6794 ax-pre-ltirr 6795 ax-pre-ltwlin 6796 ax-pre-lttrn 6797 ax-pre-apti 6798 ax-pre-ltadd 6799 |
This theorem depends on definitions: df-bi 110 df-dc 742 df-3or 885 df-3an 886 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-nel 2204 df-ral 2305 df-rex 2306 df-reu 2307 df-rab 2309 df-v 2553 df-sbc 2759 df-csb 2847 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-nul 3219 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-int 3607 df-iun 3650 df-br 3756 df-opab 3810 df-mpt 3811 df-tr 3846 df-eprel 4017 df-id 4021 df-po 4024 df-iso 4025 df-iord 4069 df-on 4071 df-suc 4074 df-iom 4257 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-iota 4810 df-fun 4847 df-fn 4848 df-f 4849 df-f1 4850 df-fo 4851 df-f1o 4852 df-fv 4853 df-riota 5411 df-ov 5458 df-oprab 5459 df-mpt2 5460 df-1st 5709 df-2nd 5710 df-recs 5861 df-irdg 5897 df-frec 5918 df-1o 5940 df-2o 5941 df-oadd 5944 df-omul 5945 df-er 6042 df-ec 6044 df-qs 6048 df-en 6158 df-ni 6288 df-pli 6289 df-mi 6290 df-lti 6291 df-plpq 6328 df-mpq 6329 df-enq 6331 df-nqqs 6332 df-plqqs 6333 df-mqqs 6334 df-1nqqs 6335 df-rq 6336 df-ltnqqs 6337 df-enq0 6407 df-nq0 6408 df-0nq0 6409 df-plq0 6410 df-mq0 6411 df-inp 6449 df-i1p 6450 df-iplp 6451 df-iltp 6453 df-enr 6654 df-nr 6655 df-ltr 6658 df-0r 6659 df-1r 6660 df-0 6718 df-1 6719 df-r 6721 df-lt 6724 df-pnf 6859 df-mnf 6860 df-xr 6861 df-ltxr 6862 df-le 6863 df-sub 6981 df-neg 6982 df-inn 7696 df-n0 7958 df-z 8022 df-uz 8250 df-fz 8645 |
This theorem is referenced by: fzfig 8887 |
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