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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 9209 | . . . . . . . . 9 | |
2 | 1 | leidd 8426 | . . . . . . . 8 |
3 | 2 | adantr 274 | . . . . . . 7 |
4 | breq1 3990 | . . . . . . . . 9 | |
5 | breq1 3990 | . . . . . . . . 9 | |
6 | 4, 5 | bibi12d 234 | . . . . . . . 8 |
7 | 6 | rspcva 2832 | . . . . . . 7 |
8 | 3, 7 | mpbid 146 | . . . . . 6 |
9 | 8 | adantlr 474 | . . . . 5 |
10 | zre 9209 | . . . . . . . . 9 | |
11 | 10 | leidd 8426 | . . . . . . . 8 |
12 | 11 | adantr 274 | . . . . . . 7 |
13 | breq1 3990 | . . . . . . . . 9 | |
14 | breq1 3990 | . . . . . . . . 9 | |
15 | 13, 14 | bibi12d 234 | . . . . . . . 8 |
16 | 15 | rspcva 2832 | . . . . . . 7 |
17 | 12, 16 | mpbird 166 | . . . . . 6 |
18 | 17 | adantll 473 | . . . . 5 |
19 | 9, 18 | jca 304 | . . . 4 |
20 | 19 | ex 114 | . . 3 |
21 | letri3 7993 | . . . 4 | |
22 | 1, 10, 21 | syl2an 287 | . . 3 |
23 | 20, 22 | sylibrd 168 | . 2 |
24 | 23 | 3impia 1195 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 973 wceq 1348 wcel 2141 wral 2448 class class class wbr 3987 cr 7766 cle 7948 cz 9205 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-cnex 7858 ax-resscn 7859 ax-pre-ltirr 7879 ax-pre-apti 7882 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-xp 4615 df-cnv 4617 df-iota 5158 df-fv 5204 df-ov 5854 df-pnf 7949 df-mnf 7950 df-xr 7951 df-ltxr 7952 df-le 7953 df-neg 8086 df-z 9206 |
This theorem is referenced by: zextlt 9297 |
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