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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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltrelpr 7306 | . . . 4 | |
2 | 1 | brel 4586 | . . 3 |
3 | ltdfpr 7307 | . . . 4 | |
4 | 3 | biimpd 143 | . . 3 |
5 | 2, 4 | mpcom 36 | . 2 |
6 | simpll 518 | . . . . . 6 | |
7 | simpr 109 | . . . . . 6 | |
8 | simprrl 528 | . . . . . . 7 | |
9 | 8 | adantr 274 | . . . . . 6 |
10 | 2 | simpld 111 | . . . . . . . 8 |
11 | prop 7276 | . . . . . . . 8 | |
12 | 10, 11 | syl 14 | . . . . . . 7 |
13 | prltlu 7288 | . . . . . . 7 | |
14 | 12, 13 | syl3an1 1249 | . . . . . 6 |
15 | 6, 7, 9, 14 | syl3anc 1216 | . . . . 5 |
16 | simprrr 529 | . . . . . . 7 | |
17 | 16 | adantr 274 | . . . . . 6 |
18 | 2 | simprd 113 | . . . . . . . 8 |
19 | prop 7276 | . . . . . . . 8 | |
20 | 18, 19 | syl 14 | . . . . . . 7 |
21 | prcdnql 7285 | . . . . . . 7 | |
22 | 20, 21 | sylan 281 | . . . . . 6 |
23 | 6, 17, 22 | syl2anc 408 | . . . . 5 |
24 | 15, 23 | mpd 13 | . . . 4 |
25 | 24 | ex 114 | . . 3 |
26 | 25 | ssrdv 3098 | . 2 |
27 | 5, 26 | rexlimddv 2552 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wcel 1480 wrex 2415 wss 3066 cop 3525 class class class wbr 3924 cfv 5118 c1st 6029 c2nd 6030 cnq 7081 cltq 7086 cnp 7092 cltp 7096 |
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 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 |
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 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-eprel 4206 df-id 4210 df-po 4213 df-iso 4214 df-iord 4283 df-on 4285 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-irdg 6260 df-oadd 6310 df-omul 6311 df-er 6422 df-ec 6424 df-qs 6428 df-ni 7105 df-mi 7107 df-lti 7108 df-enq 7148 df-nqqs 7149 df-ltnqqs 7154 df-inp 7267 df-iltp 7271 |
This theorem is referenced by: ltexprlemrl 7411 |
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