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Mirrors > Home > ILE Home > Th. List > xporderlem | Unicode version |
Description: Lemma for lexicographical ordering theorems. (Contributed by Scott Fenton, 16-Mar-2011.) |
Ref | Expression |
---|---|
xporderlem.1 |
Ref | Expression |
---|---|
xporderlem |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-br 3765 | . . 3 | |
2 | xporderlem.1 | . . . 4 | |
3 | 2 | eleq2i 2104 | . . 3 |
4 | 1, 3 | bitri 173 | . 2 |
5 | vex 2560 | . . . 4 | |
6 | vex 2560 | . . . 4 | |
7 | 5, 6 | opex 3966 | . . 3 |
8 | vex 2560 | . . . 4 | |
9 | vex 2560 | . . . 4 | |
10 | 8, 9 | opex 3966 | . . 3 |
11 | eleq1 2100 | . . . . . 6 | |
12 | opelxp 4374 | . . . . . 6 | |
13 | 11, 12 | syl6bb 185 | . . . . 5 |
14 | 13 | anbi1d 438 | . . . 4 |
15 | 5, 6 | op1std 5775 | . . . . . 6 |
16 | 15 | breq1d 3774 | . . . . 5 |
17 | 15 | eqeq1d 2048 | . . . . . 6 |
18 | 5, 6 | op2ndd 5776 | . . . . . . 7 |
19 | 18 | breq1d 3774 | . . . . . 6 |
20 | 17, 19 | anbi12d 442 | . . . . 5 |
21 | 16, 20 | orbi12d 707 | . . . 4 |
22 | 14, 21 | anbi12d 442 | . . 3 |
23 | eleq1 2100 | . . . . . 6 | |
24 | opelxp 4374 | . . . . . 6 | |
25 | 23, 24 | syl6bb 185 | . . . . 5 |
26 | 25 | anbi2d 437 | . . . 4 |
27 | 8, 9 | op1std 5775 | . . . . . 6 |
28 | 27 | breq2d 3776 | . . . . 5 |
29 | 27 | eqeq2d 2051 | . . . . . 6 |
30 | 8, 9 | op2ndd 5776 | . . . . . . 7 |
31 | 30 | breq2d 3776 | . . . . . 6 |
32 | 29, 31 | anbi12d 442 | . . . . 5 |
33 | 28, 32 | orbi12d 707 | . . . 4 |
34 | 26, 33 | anbi12d 442 | . . 3 |
35 | 7, 10, 22, 34 | opelopab 4008 | . 2 |
36 | an4 520 | . . 3 | |
37 | 36 | anbi1i 431 | . 2 |
38 | 4, 35, 37 | 3bitri 195 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 97 wb 98 wo 629 wceq 1243 wcel 1393 cop 3378 class class class wbr 3764 copab 3817 cxp 4343 cfv 4902 c1st 5765 c2nd 5766 |
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-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 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 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-ral 2311 df-rex 2312 df-v 2559 df-sbc 2765 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-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-iota 4867 df-fun 4904 df-fv 4910 df-1st 5767 df-2nd 5768 |
This theorem is referenced by: poxp 5853 |
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