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Mirrors > Home > ILE Home > Th. List > ltprordil | Unicode version |
Description: If a positive real is less than a second positive real, its lower cut is a subset of the second's lower cut. (Contributed by Jim Kingdon, 23-Dec-2019.) |
Ref | Expression |
---|---|
ltprordil |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltrelpr 7523 |
. . . 4
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2 | 1 | brel 4693 |
. . 3
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3 | ltdfpr 7524 |
. . . 4
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4 | 3 | biimpd 144 |
. . 3
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5 | 2, 4 | mpcom 36 |
. 2
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6 | simpll 527 |
. . . . . 6
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7 | simpr 110 |
. . . . . 6
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8 | simprrl 539 |
. . . . . . 7
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9 | 8 | adantr 276 |
. . . . . 6
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10 | 2 | simpld 112 |
. . . . . . . 8
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11 | prop 7493 |
. . . . . . . 8
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12 | 10, 11 | syl 14 |
. . . . . . 7
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13 | prltlu 7505 |
. . . . . . 7
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14 | 12, 13 | syl3an1 1282 |
. . . . . 6
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15 | 6, 7, 9, 14 | syl3anc 1249 |
. . . . 5
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16 | simprrr 540 |
. . . . . . 7
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17 | 16 | adantr 276 |
. . . . . 6
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18 | 2 | simprd 114 |
. . . . . . . 8
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19 | prop 7493 |
. . . . . . . 8
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20 | 18, 19 | syl 14 |
. . . . . . 7
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21 | prcdnql 7502 |
. . . . . . 7
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22 | 20, 21 | sylan 283 |
. . . . . 6
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23 | 6, 17, 22 | syl2anc 411 |
. . . . 5
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24 | 15, 23 | mpd 13 |
. . . 4
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25 | 24 | ex 115 |
. . 3
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26 | 25 | ssrdv 3176 |
. 2
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27 | 5, 26 | rexlimddv 2612 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-iinf 4602 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-eprel 4304 df-id 4308 df-po 4311 df-iso 4312 df-iord 4381 df-on 4383 df-suc 4386 df-iom 4605 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5233 df-fn 5234 df-f 5235 df-f1 5236 df-fo 5237 df-f1o 5238 df-fv 5239 df-ov 5894 df-oprab 5895 df-mpo 5896 df-1st 6159 df-2nd 6160 df-recs 6324 df-irdg 6389 df-oadd 6439 df-omul 6440 df-er 6553 df-ec 6555 df-qs 6559 df-ni 7322 df-mi 7324 df-lti 7325 df-enq 7365 df-nqqs 7366 df-ltnqqs 7371 df-inp 7484 df-iltp 7488 |
This theorem is referenced by: ltexprlemrl 7628 |
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