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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 3999 | . . 3 | |
2 | xporderlem.1 | . . . 4 | |
3 | 2 | eleq2i 2242 | . . 3 |
4 | 1, 3 | bitri 184 | . 2 |
5 | vex 2738 | . . . 4 | |
6 | vex 2738 | . . . 4 | |
7 | 5, 6 | opex 4223 | . . 3 |
8 | vex 2738 | . . . 4 | |
9 | vex 2738 | . . . 4 | |
10 | 8, 9 | opex 4223 | . . 3 |
11 | eleq1 2238 | . . . . . 6 | |
12 | opelxp 4650 | . . . . . 6 | |
13 | 11, 12 | bitrdi 196 | . . . . 5 |
14 | 13 | anbi1d 465 | . . . 4 |
15 | 5, 6 | op1std 6139 | . . . . . 6 |
16 | 15 | breq1d 4008 | . . . . 5 |
17 | 15 | eqeq1d 2184 | . . . . . 6 |
18 | 5, 6 | op2ndd 6140 | . . . . . . 7 |
19 | 18 | breq1d 4008 | . . . . . 6 |
20 | 17, 19 | anbi12d 473 | . . . . 5 |
21 | 16, 20 | orbi12d 793 | . . . 4 |
22 | 14, 21 | anbi12d 473 | . . 3 |
23 | eleq1 2238 | . . . . . 6 | |
24 | opelxp 4650 | . . . . . 6 | |
25 | 23, 24 | bitrdi 196 | . . . . 5 |
26 | 25 | anbi2d 464 | . . . 4 |
27 | 8, 9 | op1std 6139 | . . . . . 6 |
28 | 27 | breq2d 4010 | . . . . 5 |
29 | 27 | eqeq2d 2187 | . . . . . 6 |
30 | 8, 9 | op2ndd 6140 | . . . . . . 7 |
31 | 30 | breq2d 4010 | . . . . . 6 |
32 | 29, 31 | anbi12d 473 | . . . . 5 |
33 | 28, 32 | orbi12d 793 | . . . 4 |
34 | 26, 33 | anbi12d 473 | . . 3 |
35 | 7, 10, 22, 34 | opelopab 4265 | . 2 |
36 | an4 586 | . . 3 | |
37 | 36 | anbi1i 458 | . 2 |
38 | 4, 35, 37 | 3bitri 206 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 104 wb 105 wo 708 wceq 1353 wcel 2146 cop 3592 class class class wbr 3998 copab 4058 cxp 4618 cfv 5208 c1st 6129 c2nd 6130 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-sep 4116 ax-pow 4169 ax-pr 4203 ax-un 4427 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ral 2458 df-rex 2459 df-v 2737 df-sbc 2961 df-un 3131 df-in 3133 df-ss 3140 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-br 3999 df-opab 4060 df-mpt 4061 df-id 4287 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-iota 5170 df-fun 5210 df-fv 5216 df-1st 6131 df-2nd 6132 |
This theorem is referenced by: poxp 6223 |
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