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Mirrors > Home > ILE Home > Th. List > uzennn | Unicode version |
Description: An upper integer set is equinumerous to the set of natural numbers. (Contributed by Jim Kingdon, 30-Jul-2023.) |
Ref | Expression |
---|---|
uzennn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-uz 9458 | . . . . 5 | |
2 | zex 9191 | . . . . . 6 | |
3 | 2 | mptex 5705 | . . . . 5 |
4 | 1, 3 | eqeltri 2237 | . . . 4 |
5 | fvexg 5499 | . . . 4 | |
6 | 4, 5 | mpan 421 | . . 3 |
7 | nn0ex 9111 | . . . 4 | |
8 | 7 | a1i 9 | . . 3 |
9 | eluzelz 9466 | . . . . . . 7 | |
10 | 9 | adantl 275 | . . . . . 6 |
11 | simpl 108 | . . . . . 6 | |
12 | 10, 11 | zsubcld 9309 | . . . . 5 |
13 | eluzle 9469 | . . . . . . 7 | |
14 | 13 | adantl 275 | . . . . . 6 |
15 | 10 | zred 9304 | . . . . . . 7 |
16 | 11 | zred 9304 | . . . . . . 7 |
17 | 15, 16 | subge0d 8424 | . . . . . 6 |
18 | 14, 17 | mpbird 166 | . . . . 5 |
19 | elnn0z 9195 | . . . . 5 | |
20 | 12, 18, 19 | sylanbrc 414 | . . . 4 |
21 | 20 | ex 114 | . . 3 |
22 | simpl 108 | . . . . 5 | |
23 | nn0z 9202 | . . . . . . 7 | |
24 | 23 | adantl 275 | . . . . . 6 |
25 | 24, 22 | zaddcld 9308 | . . . . 5 |
26 | nn0ge0 9130 | . . . . . . 7 | |
27 | 26 | adantl 275 | . . . . . 6 |
28 | 22 | zred 9304 | . . . . . . 7 |
29 | 24 | zred 9304 | . . . . . . 7 |
30 | 28, 29 | addge02d 8423 | . . . . . 6 |
31 | 27, 30 | mpbid 146 | . . . . 5 |
32 | eluz2 9463 | . . . . 5 | |
33 | 22, 25, 31, 32 | syl3anbrc 1170 | . . . 4 |
34 | 33 | ex 114 | . . 3 |
35 | 9 | ad2antrl 482 | . . . . . . 7 |
36 | 35 | zcnd 9305 | . . . . . 6 |
37 | simpl 108 | . . . . . . 7 | |
38 | 37 | zcnd 9305 | . . . . . 6 |
39 | simprr 522 | . . . . . . 7 | |
40 | 39 | nn0cnd 9160 | . . . . . 6 |
41 | 36, 38, 40 | subadd2d 8219 | . . . . 5 |
42 | bicom 139 | . . . . . 6 | |
43 | eqcom 2166 | . . . . . . 7 | |
44 | eqcom 2166 | . . . . . . 7 | |
45 | 43, 44 | bibi12i 228 | . . . . . 6 |
46 | 42, 45 | bitri 183 | . . . . 5 |
47 | 41, 46 | sylib 121 | . . . 4 |
48 | 47 | ex 114 | . . 3 |
49 | 6, 8, 21, 34, 48 | en3d 6726 | . 2 |
50 | nn0ennn 10358 | . 2 | |
51 | entr 6741 | . 2 | |
52 | 49, 50, 51 | sylancl 410 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1342 wcel 2135 crab 2446 cvv 2721 class class class wbr 3976 cmpt 4037 cfv 5182 (class class class)co 5836 cen 6695 cc0 7744 caddc 7747 cle 7925 cmin 8060 cn 8848 cn0 9105 cz 9182 cuz 9457 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-addass 7846 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-0id 7852 ax-rnegex 7853 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-ltadd 7860 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-er 6492 df-en 6698 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-inn 8849 df-n0 9106 df-z 9183 df-uz 9458 |
This theorem is referenced by: exmidunben 12302 |
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