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Mirrors > Home > ILE Home > Th. List > prubl | Unicode version |
Description: A positive fraction not in a lower cut is an upper bound. (Contributed by Jim Kingdon, 29-Sep-2019.) |
Ref | Expression |
---|---|
prubl |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2145 |
. . . . . . 7
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2 | 1 | biimpcd 157 |
. . . . . 6
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3 | 2 | adantl 271 |
. . . . 5
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4 | prcdnql 6813 |
. . . . 5
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5 | 3, 4 | jaod 670 |
. . . 4
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6 | 5 | con3d 594 |
. . 3
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7 | 6 | adantr 270 |
. 2
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8 | elprnql 6810 |
. . 3
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9 | nqtric 6728 |
. . 3
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10 | 8, 9 | sylan 277 |
. 2
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11 | 7, 10 | sylibrd 167 |
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 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-coll 3914 ax-sep 3917 ax-nul 3925 ax-pow 3969 ax-pr 3993 ax-un 4217 ax-setind 4309 ax-iinf 4358 |
This theorem depends on definitions: df-bi 115 df-dc 777 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-ral 2358 df-rex 2359 df-reu 2360 df-rab 2362 df-v 2613 df-sbc 2826 df-csb 2919 df-dif 2985 df-un 2987 df-in 2989 df-ss 2996 df-nul 3269 df-pw 3403 df-sn 3423 df-pr 3424 df-op 3426 df-uni 3623 df-int 3658 df-iun 3701 df-br 3807 df-opab 3861 df-mpt 3862 df-tr 3897 df-eprel 4073 df-id 4077 df-po 4080 df-iso 4081 df-iord 4150 df-on 4152 df-suc 4155 df-iom 4361 df-xp 4398 df-rel 4399 df-cnv 4400 df-co 4401 df-dm 4402 df-rn 4403 df-res 4404 df-ima 4405 df-iota 4918 df-fun 4955 df-fn 4956 df-f 4957 df-f1 4958 df-fo 4959 df-f1o 4960 df-fv 4961 df-ov 5568 df-oprab 5569 df-mpt2 5570 df-1st 5820 df-2nd 5821 df-recs 5976 df-irdg 6041 df-oadd 6091 df-omul 6092 df-er 6195 df-ec 6197 df-qs 6201 df-ni 6633 df-mi 6635 df-lti 6636 df-enq 6676 df-nqqs 6677 df-ltnqqs 6682 df-inp 6795 |
This theorem is referenced by: prltlu 6816 |
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