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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 9333 | . . . . 5 | |
2 | eleq2 2201 | . . . . . 6 | |
3 | eluzel2 9324 | . . . . . 6 | |
4 | 2, 3 | syl6bi 162 | . . . . 5 |
5 | 1, 4 | mpan9 279 | . . . 4 |
6 | uzid 9333 | . . . . . . . . . . 11 | |
7 | eleq2 2201 | . . . . . . . . . . 11 | |
8 | 6, 7 | syl5ibr 155 | . . . . . . . . . 10 |
9 | eluzle 9331 | . . . . . . . . . 10 | |
10 | 8, 9 | syl6 33 | . . . . . . . . 9 |
11 | 1, 2 | syl5ib 153 | . . . . . . . . . 10 |
12 | eluzle 9331 | . . . . . . . . . 10 | |
13 | 11, 12 | syl6 33 | . . . . . . . . 9 |
14 | 10, 13 | anim12d 333 | . . . . . . . 8 |
15 | 14 | impl 377 | . . . . . . 7 |
16 | 15 | ancoms 266 | . . . . . 6 |
17 | 16 | anassrs 397 | . . . . 5 |
18 | zre 9051 | . . . . . . 7 | |
19 | zre 9051 | . . . . . . 7 | |
20 | letri3 7838 | . . . . . . 7 | |
21 | 18, 19, 20 | syl2an 287 | . . . . . 6 |
22 | 21 | adantlr 468 | . . . . 5 |
23 | 17, 22 | mpbird 166 | . . . 4 |
24 | 5, 23 | mpdan 417 | . . 3 |
25 | 24 | ex 114 | . 2 |
26 | fveq2 5414 | . 2 | |
27 | 25, 26 | impbid1 141 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 class class class wbr 3924 cfv 5118 cr 7612 cle 7794 cz 9047 cuz 9319 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-pre-ltirr 7725 ax-pre-apti 7728 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-fv 5126 df-ov 5770 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-neg 7929 df-z 9048 df-uz 9320 |
This theorem is referenced by: fzopth 9834 |
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