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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 | frec |
Ref | Expression |
---|---|
frecfzen2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluzel2 9479 | . . . 4 | |
2 | eluzelz 9483 | . . . 4 | |
3 | 1z 9225 | . . . . 5 | |
4 | zsubcl 9240 | . . . . 5 | |
5 | 3, 1, 4 | sylancr 412 | . . . 4 |
6 | fzen 9986 | . . . 4 | |
7 | 1, 2, 5, 6 | syl3anc 1233 | . . 3 |
8 | 1 | zcnd 9322 | . . . . 5 |
9 | ax-1cn 7854 | . . . . 5 | |
10 | pncan3 8114 | . . . . 5 | |
11 | 8, 9, 10 | sylancl 411 | . . . 4 |
12 | zcn 9204 | . . . . . . 7 | |
13 | zcn 9204 | . . . . . . 7 | |
14 | addsubass 8116 | . . . . . . . 8 | |
15 | 9, 14 | mp3an2 1320 | . . . . . . 7 |
16 | 12, 13, 15 | syl2an 287 | . . . . . 6 |
17 | 2, 1, 16 | syl2anc 409 | . . . . 5 |
18 | 17 | eqcomd 2176 | . . . 4 |
19 | 11, 18 | oveq12d 5868 | . . 3 |
20 | 7, 19 | breqtrd 4013 | . 2 |
21 | peano2uz 9529 | . . 3 | |
22 | uznn0sub 9505 | . . 3 | |
23 | frecfzennn.1 | . . . 4 frec | |
24 | 23 | frecfzennn 10369 | . . 3 |
25 | 21, 22, 24 | 3syl 17 | . 2 |
26 | entr 6758 | . 2 | |
27 | 20, 25, 26 | syl2anc 409 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1348 wcel 2141 class class class wbr 3987 cmpt 4048 ccnv 4608 cfv 5196 (class class class)co 5850 freccfrec 6366 cen 6712 cc 7759 cc0 7761 c1 7762 caddc 7764 cmin 8077 cn0 9122 cz 9199 cuz 9474 cfz 9952 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4102 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-iinf 4570 ax-cnex 7852 ax-resscn 7853 ax-1cn 7854 ax-1re 7855 ax-icn 7856 ax-addcl 7857 ax-addrcl 7858 ax-mulcl 7859 ax-addcom 7861 ax-addass 7863 ax-distr 7865 ax-i2m1 7866 ax-0lt1 7867 ax-0id 7869 ax-rnegex 7870 ax-cnre 7872 ax-pre-ltirr 7873 ax-pre-ltwlin 7874 ax-pre-lttrn 7875 ax-pre-apti 7876 ax-pre-ltadd 7877 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-tr 4086 df-id 4276 df-iord 4349 df-on 4351 df-ilim 4352 df-suc 4354 df-iom 4573 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-riota 5806 df-ov 5853 df-oprab 5854 df-mpo 5855 df-1st 6116 df-2nd 6117 df-recs 6281 df-frec 6367 df-1o 6392 df-er 6509 df-en 6715 df-pnf 7943 df-mnf 7944 df-xr 7945 df-ltxr 7946 df-le 7947 df-sub 8079 df-neg 8080 df-inn 8866 df-n0 9123 df-z 9200 df-uz 9475 df-fz 9953 |
This theorem is referenced by: fzfig 10373 |
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