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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 9051 | . . . . . . . . 9 | |
2 | 1 | leidd 8269 | . . . . . . . 8 |
3 | 2 | adantr 274 | . . . . . . 7 |
4 | breq1 3927 | . . . . . . . . 9 | |
5 | breq1 3927 | . . . . . . . . 9 | |
6 | 4, 5 | bibi12d 234 | . . . . . . . 8 |
7 | 6 | rspcva 2782 | . . . . . . 7 |
8 | 3, 7 | mpbid 146 | . . . . . 6 |
9 | 8 | adantlr 468 | . . . . 5 |
10 | zre 9051 | . . . . . . . . 9 | |
11 | 10 | leidd 8269 | . . . . . . . 8 |
12 | 11 | adantr 274 | . . . . . . 7 |
13 | breq1 3927 | . . . . . . . . 9 | |
14 | breq1 3927 | . . . . . . . . 9 | |
15 | 13, 14 | bibi12d 234 | . . . . . . . 8 |
16 | 15 | rspcva 2782 | . . . . . . 7 |
17 | 12, 16 | mpbird 166 | . . . . . 6 |
18 | 17 | adantll 467 | . . . . 5 |
19 | 9, 18 | jca 304 | . . . 4 |
20 | 19 | ex 114 | . . 3 |
21 | letri3 7838 | . . . 4 | |
22 | 1, 10, 21 | syl2an 287 | . . 3 |
23 | 20, 22 | sylibrd 168 | . 2 |
24 | 23 | 3impia 1178 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 962 wceq 1331 wcel 1480 wral 2414 class class class wbr 3924 cr 7612 cle 7794 cz 9047 |
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-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-xp 4540 df-cnv 4542 df-iota 5083 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 |
This theorem is referenced by: zextlt 9136 |
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