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Mirrors > Home > ILE Home > Th. List > genplt2i | Unicode version |
Description: Operating on both sides
of two inequalities, when the operation is
consistent with ![]() |
Ref | Expression |
---|---|
genplt2i.ord |
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genplt2i.com |
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Ref | Expression |
---|---|
genplt2i |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 107 |
. . 3
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2 | genplt2i.ord |
. . . . 5
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3 | 2 | adantl 271 |
. . . 4
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4 | ltrelnq 6914 |
. . . . . 6
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5 | 4 | brel 4486 |
. . . . 5
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6 | 4 | brel 4486 |
. . . . 5
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7 | simpll 496 |
. . . . 5
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8 | 5, 6, 7 | syl2an 283 |
. . . 4
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9 | simplr 497 |
. . . . 5
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10 | 5, 6, 9 | syl2an 283 |
. . . 4
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11 | simprl 498 |
. . . . 5
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12 | 5, 6, 11 | syl2an 283 |
. . . 4
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13 | genplt2i.com |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
14 | 13 | adantl 271 |
. . . 4
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15 | 3, 8, 10, 12, 14 | caovord2d 5806 |
. . 3
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16 | 1, 15 | mpbid 145 |
. 2
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17 | simpr 108 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
18 | simprr 499 |
. . . . 5
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19 | 5, 6, 18 | syl2an 283 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
20 | 3, 12, 19, 10 | caovordd 5805 |
. . 3
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21 | 17, 20 | mpbid 145 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
22 | ltsonq 6947 |
. . 3
![]() ![]() ![]() ![]() | |
23 | 22, 4 | sotri 4822 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | 16, 21, 23 | syl2anc 403 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-coll 3952 ax-sep 3955 ax-nul 3963 ax-pow 4007 ax-pr 4034 ax-un 4258 ax-setind 4351 ax-iinf 4401 |
This theorem depends on definitions: df-bi 115 df-dc 781 df-3or 925 df-3an 926 df-tru 1292 df-fal 1295 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ne 2256 df-ral 2364 df-rex 2365 df-reu 2366 df-rab 2368 df-v 2621 df-sbc 2841 df-csb 2934 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-nul 3287 df-pw 3429 df-sn 3450 df-pr 3451 df-op 3453 df-uni 3652 df-int 3687 df-iun 3730 df-br 3844 df-opab 3898 df-mpt 3899 df-tr 3935 df-eprel 4114 df-id 4118 df-po 4121 df-iso 4122 df-iord 4191 df-on 4193 df-suc 4196 df-iom 4404 df-xp 4442 df-rel 4443 df-cnv 4444 df-co 4445 df-dm 4446 df-rn 4447 df-res 4448 df-ima 4449 df-iota 4975 df-fun 5012 df-fn 5013 df-f 5014 df-f1 5015 df-fo 5016 df-f1o 5017 df-fv 5018 df-ov 5647 df-oprab 5648 df-mpt2 5649 df-1st 5903 df-2nd 5904 df-recs 6062 df-irdg 6127 df-oadd 6177 df-omul 6178 df-er 6282 df-ec 6284 df-qs 6288 df-ni 6853 df-mi 6855 df-lti 6856 df-enq 6896 df-nqqs 6897 df-ltnqqs 6902 |
This theorem is referenced by: genprndl 7070 genprndu 7071 genpdisj 7072 |
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