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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 3925 | . . 3 | |
2 | xporderlem.1 | . . . 4 | |
3 | 2 | eleq2i 2204 | . . 3 |
4 | 1, 3 | bitri 183 | . 2 |
5 | vex 2684 | . . . 4 | |
6 | vex 2684 | . . . 4 | |
7 | 5, 6 | opex 4146 | . . 3 |
8 | vex 2684 | . . . 4 | |
9 | vex 2684 | . . . 4 | |
10 | 8, 9 | opex 4146 | . . 3 |
11 | eleq1 2200 | . . . . . 6 | |
12 | opelxp 4564 | . . . . . 6 | |
13 | 11, 12 | syl6bb 195 | . . . . 5 |
14 | 13 | anbi1d 460 | . . . 4 |
15 | 5, 6 | op1std 6039 | . . . . . 6 |
16 | 15 | breq1d 3934 | . . . . 5 |
17 | 15 | eqeq1d 2146 | . . . . . 6 |
18 | 5, 6 | op2ndd 6040 | . . . . . . 7 |
19 | 18 | breq1d 3934 | . . . . . 6 |
20 | 17, 19 | anbi12d 464 | . . . . 5 |
21 | 16, 20 | orbi12d 782 | . . . 4 |
22 | 14, 21 | anbi12d 464 | . . 3 |
23 | eleq1 2200 | . . . . . 6 | |
24 | opelxp 4564 | . . . . . 6 | |
25 | 23, 24 | syl6bb 195 | . . . . 5 |
26 | 25 | anbi2d 459 | . . . 4 |
27 | 8, 9 | op1std 6039 | . . . . . 6 |
28 | 27 | breq2d 3936 | . . . . 5 |
29 | 27 | eqeq2d 2149 | . . . . . 6 |
30 | 8, 9 | op2ndd 6040 | . . . . . . 7 |
31 | 30 | breq2d 3936 | . . . . . 6 |
32 | 29, 31 | anbi12d 464 | . . . . 5 |
33 | 28, 32 | orbi12d 782 | . . . 4 |
34 | 26, 33 | anbi12d 464 | . . 3 |
35 | 7, 10, 22, 34 | opelopab 4188 | . 2 |
36 | an4 575 | . . 3 | |
37 | 36 | anbi1i 453 | . 2 |
38 | 4, 35, 37 | 3bitri 205 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wo 697 wceq 1331 wcel 1480 cop 3525 class class class wbr 3924 copab 3983 cxp 4532 cfv 5118 c1st 6029 c2nd 6030 |
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-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 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 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-ral 2419 df-rex 2420 df-v 2683 df-sbc 2905 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-iota 5083 df-fun 5120 df-fv 5126 df-1st 6031 df-2nd 6032 |
This theorem is referenced by: poxp 6122 |
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