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Mirrors > Home > ILE Home > Th. List > zextle | Unicode version |
Description: An extensionality-like property for integer ordering. (Contributed by NM, 29-Oct-2005.) |
Ref | Expression |
---|---|
zextle |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zre 8249 | . . . . . . . . 9 | |
2 | 1 | leidd 7506 | . . . . . . . 8 |
3 | 2 | adantr 261 | . . . . . . 7 |
4 | breq1 3767 | . . . . . . . . 9 | |
5 | breq1 3767 | . . . . . . . . 9 | |
6 | 4, 5 | bibi12d 224 | . . . . . . . 8 |
7 | 6 | rspcva 2654 | . . . . . . 7 |
8 | 3, 7 | mpbid 135 | . . . . . 6 |
9 | 8 | adantlr 446 | . . . . 5 |
10 | zre 8249 | . . . . . . . . 9 | |
11 | 10 | leidd 7506 | . . . . . . . 8 |
12 | 11 | adantr 261 | . . . . . . 7 |
13 | breq1 3767 | . . . . . . . . 9 | |
14 | breq1 3767 | . . . . . . . . 9 | |
15 | 13, 14 | bibi12d 224 | . . . . . . . 8 |
16 | 15 | rspcva 2654 | . . . . . . 7 |
17 | 12, 16 | mpbird 156 | . . . . . 6 |
18 | 17 | adantll 445 | . . . . 5 |
19 | 9, 18 | jca 290 | . . . 4 |
20 | 19 | ex 108 | . . 3 |
21 | letri3 7099 | . . . 4 | |
22 | 1, 10, 21 | syl2an 273 | . . 3 |
23 | 20, 22 | sylibrd 158 | . 2 |
24 | 23 | 3impia 1101 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 w3a 885 wceq 1243 wcel 1393 wral 2306 class class class wbr 3764 cr 6888 cle 7061 cz 8245 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-cnex 6975 ax-resscn 6976 ax-pre-ltirr 6996 ax-pre-apti 6999 |
This theorem depends on definitions: df-bi 110 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-nel 2207 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-xp 4351 df-cnv 4353 df-iota 4867 df-fv 4910 df-ov 5515 df-pnf 7062 df-mnf 7063 df-xr 7064 df-ltxr 7065 df-le 7066 df-neg 7185 df-z 8246 |
This theorem is referenced by: zextlt 8332 |
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