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Mirrors > Home > ILE Home > Th. List > prltlu | Unicode version |
Description: An element of a lower cut is less than an element of the corresponding upper cut. (Contributed by Jim Kingdon, 15-Oct-2019.) |
Ref | Expression |
---|---|
prltlu |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp3 984 | . . 3 | |
2 | eleq1 2220 | . . . . . . 7 | |
3 | eleq1 2220 | . . . . . . 7 | |
4 | 2, 3 | anbi12d 465 | . . . . . 6 |
5 | 4 | notbid 657 | . . . . 5 |
6 | elinp 7389 | . . . . . . 7 | |
7 | simpr2 989 | . . . . . . 7 | |
8 | 6, 7 | sylbi 120 | . . . . . 6 |
9 | 8 | 3ad2ant1 1003 | . . . . 5 |
10 | elprnqu 7397 | . . . . . 6 | |
11 | 10 | 3adant2 1001 | . . . . 5 |
12 | 5, 9, 11 | rspcdva 2821 | . . . 4 |
13 | ancom 264 | . . . . . 6 | |
14 | 13 | notbii 658 | . . . . 5 |
15 | imnan 680 | . . . . 5 | |
16 | 14, 15 | bitr4i 186 | . . . 4 |
17 | 12, 16 | sylib 121 | . . 3 |
18 | 1, 17 | mpd 13 | . 2 |
19 | 3simpa 979 | . . 3 | |
20 | prubl 7401 | . . 3 | |
21 | 19, 11, 20 | syl2anc 409 | . 2 |
22 | 18, 21 | mpd 13 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 698 w3a 963 wceq 1335 wcel 2128 wral 2435 wrex 2436 wss 3102 cop 3563 class class class wbr 3965 cnq 7195 cltq 7200 cnp 7206 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4079 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4495 ax-iinf 4546 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-tr 4063 df-eprel 4249 df-id 4253 df-po 4256 df-iso 4257 df-iord 4326 df-on 4328 df-suc 4331 df-iom 4549 df-xp 4591 df-rel 4592 df-cnv 4593 df-co 4594 df-dm 4595 df-rn 4596 df-res 4597 df-ima 4598 df-iota 5134 df-fun 5171 df-fn 5172 df-f 5173 df-f1 5174 df-fo 5175 df-f1o 5176 df-fv 5177 df-ov 5824 df-oprab 5825 df-mpo 5826 df-1st 6085 df-2nd 6086 df-recs 6249 df-irdg 6314 df-oadd 6364 df-omul 6365 df-er 6477 df-ec 6479 df-qs 6483 df-ni 7219 df-mi 7221 df-lti 7222 df-enq 7262 df-nqqs 7263 df-ltnqqs 7268 df-inp 7381 |
This theorem is referenced by: genpdisj 7438 prmuloc 7481 ltprordil 7504 ltpopr 7510 ltexprlemopu 7518 ltexprlemdisj 7521 ltexprlemfl 7524 ltexprlemfu 7526 ltexprlemru 7527 recexprlemdisj 7545 recexprlemss1l 7550 recexprlemss1u 7551 |
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