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Mirrors > Home > ILE Home > Th. List > fzopth | Unicode version |
Description: A finite set of sequential integers can represent an ordered pair. (Contributed by NM, 31-Oct-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
fzopth |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluzfz1 9956 | . . . . . . . . 9 | |
2 | 1 | adantr 274 | . . . . . . . 8 |
3 | simpr 109 | . . . . . . . 8 | |
4 | 2, 3 | eleqtrd 2243 | . . . . . . 7 |
5 | elfzuz 9947 | . . . . . . 7 | |
6 | uzss 9477 | . . . . . . 7 | |
7 | 4, 5, 6 | 3syl 17 | . . . . . 6 |
8 | elfzuz2 9954 | . . . . . . . . 9 | |
9 | eluzfz1 9956 | . . . . . . . . 9 | |
10 | 4, 8, 9 | 3syl 17 | . . . . . . . 8 |
11 | 10, 3 | eleqtrrd 2244 | . . . . . . 7 |
12 | elfzuz 9947 | . . . . . . 7 | |
13 | uzss 9477 | . . . . . . 7 | |
14 | 11, 12, 13 | 3syl 17 | . . . . . 6 |
15 | 7, 14 | eqssd 3154 | . . . . 5 |
16 | eluzel2 9462 | . . . . . . 7 | |
17 | 16 | adantr 274 | . . . . . 6 |
18 | uz11 9479 | . . . . . 6 | |
19 | 17, 18 | syl 14 | . . . . 5 |
20 | 15, 19 | mpbid 146 | . . . 4 |
21 | eluzfz2 9957 | . . . . . . . . 9 | |
22 | 4, 8, 21 | 3syl 17 | . . . . . . . 8 |
23 | 22, 3 | eleqtrrd 2244 | . . . . . . 7 |
24 | elfzuz3 9948 | . . . . . . 7 | |
25 | uzss 9477 | . . . . . . 7 | |
26 | 23, 24, 25 | 3syl 17 | . . . . . 6 |
27 | eluzfz2 9957 | . . . . . . . . 9 | |
28 | 27 | adantr 274 | . . . . . . . 8 |
29 | 28, 3 | eleqtrd 2243 | . . . . . . 7 |
30 | elfzuz3 9948 | . . . . . . 7 | |
31 | uzss 9477 | . . . . . . 7 | |
32 | 29, 30, 31 | 3syl 17 | . . . . . 6 |
33 | 26, 32 | eqssd 3154 | . . . . 5 |
34 | eluzelz 9466 | . . . . . . 7 | |
35 | 34 | adantr 274 | . . . . . 6 |
36 | uz11 9479 | . . . . . 6 | |
37 | 35, 36 | syl 14 | . . . . 5 |
38 | 33, 37 | mpbid 146 | . . . 4 |
39 | 20, 38 | jca 304 | . . 3 |
40 | 39 | ex 114 | . 2 |
41 | oveq12 5845 | . 2 | |
42 | 40, 41 | impbid1 141 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1342 wcel 2135 wss 3111 cfv 5182 (class class class)co 5836 cz 9182 cuz 9457 cfz 9935 |
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-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-apti 7859 |
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-rab 2451 df-v 2723 df-sbc 2947 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-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-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-neg 8063 df-z 9183 df-uz 9458 df-fz 9936 |
This theorem is referenced by: fz0to4untppr 10049 2ffzeq 10066 |
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