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Mirrors > Home > ILE Home > Th. List > ltnnnq | Unicode version |
Description: Ordering of positive integers via or is equivalent. (Contributed by Jim Kingdon, 3-Oct-2020.) |
Ref | Expression |
---|---|
ltnnnq |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 108 | . . 3 | |
2 | 1pi 7123 | . . . 4 | |
3 | 2 | a1i 9 | . . 3 |
4 | simpr 109 | . . 3 | |
5 | ordpipqqs 7182 | . . 3 | |
6 | 1, 3, 4, 3, 5 | syl22anc 1217 | . 2 |
7 | mulidpi 7126 | . . . 4 | |
8 | 1, 7 | syl 14 | . . 3 |
9 | mulcompig 7139 | . . . . 5 | |
10 | 2, 4, 9 | sylancr 410 | . . . 4 |
11 | mulidpi 7126 | . . . . 5 | |
12 | 4, 11 | syl 14 | . . . 4 |
13 | 10, 12 | eqtrd 2172 | . . 3 |
14 | 8, 13 | breq12d 3942 | . 2 |
15 | 6, 14 | bitr2d 188 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 cop 3530 class class class wbr 3929 (class class class)co 5774 c1o 6306 cec 6427 cnpi 7080 cmi 7082 clti 7083 ceq 7087 cltq 7093 |
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 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-eprel 4211 df-id 4215 df-iord 4288 df-on 4290 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-1o 6313 df-oadd 6317 df-omul 6318 df-er 6429 df-ec 6431 df-qs 6435 df-ni 7112 df-mi 7114 df-lti 7115 df-enq 7155 df-nqqs 7156 df-ltnqqs 7161 |
This theorem is referenced by: caucvgprlemk 7473 caucvgprprlemk 7491 ltrennb 7662 |
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