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Mirrors > Home > ILE Home > Th. List > uz11 | Unicode version |
Description: The upper integers function is one-to-one. (Contributed by NM, 12-Dec-2005.) |
Ref | Expression |
---|---|
uz11 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uzid 9474 | . . . . 5 | |
2 | eleq2 2228 | . . . . . 6 | |
3 | eluzel2 9465 | . . . . . 6 | |
4 | 2, 3 | syl6bi 162 | . . . . 5 |
5 | 1, 4 | mpan9 279 | . . . 4 |
6 | uzid 9474 | . . . . . . . . . . 11 | |
7 | eleq2 2228 | . . . . . . . . . . 11 | |
8 | 6, 7 | syl5ibr 155 | . . . . . . . . . 10 |
9 | eluzle 9472 | . . . . . . . . . 10 | |
10 | 8, 9 | syl6 33 | . . . . . . . . 9 |
11 | 1, 2 | syl5ib 153 | . . . . . . . . . 10 |
12 | eluzle 9472 | . . . . . . . . . 10 | |
13 | 11, 12 | syl6 33 | . . . . . . . . 9 |
14 | 10, 13 | anim12d 333 | . . . . . . . 8 |
15 | 14 | impl 378 | . . . . . . 7 |
16 | 15 | ancoms 266 | . . . . . 6 |
17 | 16 | anassrs 398 | . . . . 5 |
18 | zre 9189 | . . . . . . 7 | |
19 | zre 9189 | . . . . . . 7 | |
20 | letri3 7973 | . . . . . . 7 | |
21 | 18, 19, 20 | syl2an 287 | . . . . . 6 |
22 | 21 | adantlr 469 | . . . . 5 |
23 | 17, 22 | mpbird 166 | . . . 4 |
24 | 5, 23 | mpdan 418 | . . 3 |
25 | 24 | ex 114 | . 2 |
26 | fveq2 5483 | . 2 | |
27 | 25, 26 | impbid1 141 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1342 wcel 2135 class class class wbr 3979 cfv 5185 cr 7746 cle 7928 cz 9185 cuz 9460 |
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 4097 ax-pow 4150 ax-pr 4184 ax-un 4408 ax-setind 4511 ax-cnex 7838 ax-resscn 7839 ax-pre-ltirr 7859 ax-pre-apti 7862 |
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 2726 df-sbc 2950 df-dif 3116 df-un 3118 df-in 3120 df-ss 3127 df-pw 3558 df-sn 3579 df-pr 3580 df-op 3582 df-uni 3787 df-br 3980 df-opab 4041 df-mpt 4042 df-id 4268 df-xp 4607 df-rel 4608 df-cnv 4609 df-co 4610 df-dm 4611 df-rn 4612 df-res 4613 df-ima 4614 df-iota 5150 df-fun 5187 df-fn 5188 df-f 5189 df-fv 5193 df-ov 5842 df-pnf 7929 df-mnf 7930 df-xr 7931 df-ltxr 7932 df-le 7933 df-neg 8066 df-z 9186 df-uz 9461 |
This theorem is referenced by: fzopth 9990 |
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