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Mirrors > Home > ILE Home > Th. List > ltbtwnnqq | Unicode version |
Description: There exists a number between any two positive fractions. Proposition 9-2.6(i) of [Gleason] p. 120. (Contributed by Jim Kingdon, 24-Sep-2019.) |
Ref | Expression |
---|---|
ltbtwnnqq |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltrelnq 7297 | . . . . 5 | |
2 | 1 | brel 4650 | . . . 4 |
3 | 2 | simpld 111 | . . 3 |
4 | ltexnqi 7341 | . . 3 | |
5 | nsmallnq 7345 | . . . . . 6 | |
6 | 1 | brel 4650 | . . . . . . . . . . . . . . 15 |
7 | 6 | simpld 111 | . . . . . . . . . . . . . 14 |
8 | ltaddnq 7339 | . . . . . . . . . . . . . 14 | |
9 | 7, 8 | sylan2 284 | . . . . . . . . . . . . 13 |
10 | 9 | ancoms 266 | . . . . . . . . . . . 12 |
11 | 10 | adantr 274 | . . . . . . . . . . 11 |
12 | ltanqi 7334 | . . . . . . . . . . . . 13 | |
13 | 12 | adantr 274 | . . . . . . . . . . . 12 |
14 | breq2 3980 | . . . . . . . . . . . . 13 | |
15 | 14 | adantl 275 | . . . . . . . . . . . 12 |
16 | 13, 15 | mpbid 146 | . . . . . . . . . . 11 |
17 | addclnq 7307 | . . . . . . . . . . . . . . 15 | |
18 | 7, 17 | sylan2 284 | . . . . . . . . . . . . . 14 |
19 | 18 | ancoms 266 | . . . . . . . . . . . . 13 |
20 | 19 | adantr 274 | . . . . . . . . . . . 12 |
21 | breq2 3980 | . . . . . . . . . . . . . 14 | |
22 | breq1 3979 | . . . . . . . . . . . . . 14 | |
23 | 21, 22 | anbi12d 465 | . . . . . . . . . . . . 13 |
24 | 23 | adantl 275 | . . . . . . . . . . . 12 |
25 | 20, 24 | rspcedv 2829 | . . . . . . . . . . 11 |
26 | 11, 16, 25 | mp2and 430 | . . . . . . . . . 10 |
27 | 26 | 3impa 1183 | . . . . . . . . 9 |
28 | 27 | 3coml 1199 | . . . . . . . 8 |
29 | 28 | 3expia 1194 | . . . . . . 7 |
30 | 29 | exlimdv 1806 | . . . . . 6 |
31 | 5, 30 | syl5 32 | . . . . 5 |
32 | 31 | impancom 258 | . . . 4 |
33 | 32 | rexlimdva 2581 | . . 3 |
34 | 3, 4, 33 | sylc 62 | . 2 |
35 | ltsonq 7330 | . . . 4 | |
36 | 35, 1 | sotri 4993 | . . 3 |
37 | 36 | rexlimivw 2577 | . 2 |
38 | 34, 37 | impbii 125 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1342 wex 1479 wcel 2135 wrex 2443 class class class wbr 3976 (class class class)co 5836 cnq 7212 cplq 7214 cltq 7217 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-eprel 4261 df-id 4265 df-po 4268 df-iso 4269 df-iord 4338 df-on 4340 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-irdg 6329 df-1o 6375 df-oadd 6379 df-omul 6380 df-er 6492 df-ec 6494 df-qs 6498 df-ni 7236 df-pli 7237 df-mi 7238 df-lti 7239 df-plpq 7276 df-mpq 7277 df-enq 7279 df-nqqs 7280 df-plqqs 7281 df-mqqs 7282 df-1nqqs 7283 df-rq 7284 df-ltnqqs 7285 |
This theorem is referenced by: ltbtwnnq 7348 nqprrnd 7475 appdivnq 7495 ltnqpr 7525 ltnqpri 7526 recexprlemopl 7557 recexprlemopu 7559 cauappcvgprlemopl 7578 cauappcvgprlemopu 7580 cauappcvgprlem2 7592 caucvgprlemopl 7601 caucvgprlemopu 7603 caucvgprlem2 7612 suplocexprlemru 7651 suplocexprlemloc 7653 |
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