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Mirrors > Home > ILE Home > Th. List > qbtwnxr | Unicode version |
Description: The rational numbers are dense in : any two extended real numbers have a rational between them. (Contributed by NM, 6-Feb-2007.) (Proof shortened by Mario Carneiro, 23-Aug-2015.) |
Ref | Expression |
---|---|
qbtwnxr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxr 9703 | . . 3 | |
2 | elxr 9703 | . . . . 5 | |
3 | qbtwnre 10182 | . . . . . . 7 | |
4 | 3 | 3expia 1194 | . . . . . 6 |
5 | simpl 108 | . . . . . . . . 9 | |
6 | peano2re 8025 | . . . . . . . . . 10 | |
7 | 6 | adantr 274 | . . . . . . . . 9 |
8 | ltp1 8730 | . . . . . . . . . 10 | |
9 | 8 | adantr 274 | . . . . . . . . 9 |
10 | qbtwnre 10182 | . . . . . . . . 9 | |
11 | 5, 7, 9, 10 | syl3anc 1227 | . . . . . . . 8 |
12 | qre 9554 | . . . . . . . . . . . . . 14 | |
13 | ltpnf 9707 | . . . . . . . . . . . . . 14 | |
14 | 12, 13 | syl 14 | . . . . . . . . . . . . 13 |
15 | 14 | adantl 275 | . . . . . . . . . . . 12 |
16 | simplr 520 | . . . . . . . . . . . 12 | |
17 | 15, 16 | breqtrrd 4004 | . . . . . . . . . . 11 |
18 | 17 | a1d 22 | . . . . . . . . . 10 |
19 | 18 | anim2d 335 | . . . . . . . . 9 |
20 | 19 | reximdva 2566 | . . . . . . . 8 |
21 | 11, 20 | mpd 13 | . . . . . . 7 |
22 | 21 | a1d 22 | . . . . . 6 |
23 | rexr 7935 | . . . . . . 7 | |
24 | breq2 3980 | . . . . . . . . 9 | |
25 | 24 | adantl 275 | . . . . . . . 8 |
26 | nltmnf 9715 | . . . . . . . . . 10 | |
27 | 26 | adantr 274 | . . . . . . . . 9 |
28 | 27 | pm2.21d 609 | . . . . . . . 8 |
29 | 25, 28 | sylbid 149 | . . . . . . 7 |
30 | 23, 29 | sylan 281 | . . . . . 6 |
31 | 4, 22, 30 | 3jaodan 1295 | . . . . 5 |
32 | 2, 31 | sylan2b 285 | . . . 4 |
33 | breq1 3979 | . . . . . 6 | |
34 | 33 | adantr 274 | . . . . 5 |
35 | pnfnlt 9714 | . . . . . . 7 | |
36 | 35 | adantl 275 | . . . . . 6 |
37 | 36 | pm2.21d 609 | . . . . 5 |
38 | 34, 37 | sylbid 149 | . . . 4 |
39 | peano2rem 8156 | . . . . . . . . . 10 | |
40 | 39 | adantl 275 | . . . . . . . . 9 |
41 | simpr 109 | . . . . . . . . 9 | |
42 | ltm1 8732 | . . . . . . . . . 10 | |
43 | 42 | adantl 275 | . . . . . . . . 9 |
44 | qbtwnre 10182 | . . . . . . . . 9 | |
45 | 40, 41, 43, 44 | syl3anc 1227 | . . . . . . . 8 |
46 | simpll 519 | . . . . . . . . . . . 12 | |
47 | 12 | adantl 275 | . . . . . . . . . . . . 13 |
48 | mnflt 9710 | . . . . . . . . . . . . 13 | |
49 | 47, 48 | syl 14 | . . . . . . . . . . . 12 |
50 | 46, 49 | eqbrtrd 3998 | . . . . . . . . . . 11 |
51 | 50 | a1d 22 | . . . . . . . . . 10 |
52 | 51 | anim1d 334 | . . . . . . . . 9 |
53 | 52 | reximdva 2566 | . . . . . . . 8 |
54 | 45, 53 | mpd 13 | . . . . . . 7 |
55 | 54 | a1d 22 | . . . . . 6 |
56 | 1re 7889 | . . . . . . . . . 10 | |
57 | mnflt 9710 | . . . . . . . . . 10 | |
58 | 56, 57 | ax-mp 5 | . . . . . . . . 9 |
59 | breq1 3979 | . . . . . . . . 9 | |
60 | 58, 59 | mpbiri 167 | . . . . . . . 8 |
61 | ltpnf 9707 | . . . . . . . . . 10 | |
62 | 56, 61 | ax-mp 5 | . . . . . . . . 9 |
63 | breq2 3980 | . . . . . . . . 9 | |
64 | 62, 63 | mpbiri 167 | . . . . . . . 8 |
65 | 1z 9208 | . . . . . . . . . 10 | |
66 | zq 9555 | . . . . . . . . . 10 | |
67 | 65, 66 | ax-mp 5 | . . . . . . . . 9 |
68 | breq2 3980 | . . . . . . . . . . 11 | |
69 | breq1 3979 | . . . . . . . . . . 11 | |
70 | 68, 69 | anbi12d 465 | . . . . . . . . . 10 |
71 | 70 | rspcev 2825 | . . . . . . . . 9 |
72 | 67, 71 | mpan 421 | . . . . . . . 8 |
73 | 60, 64, 72 | syl2an 287 | . . . . . . 7 |
74 | 73 | a1d 22 | . . . . . 6 |
75 | 3mix3 1157 | . . . . . . . 8 | |
76 | 75, 1 | sylibr 133 | . . . . . . 7 |
77 | 76, 29 | sylan 281 | . . . . . 6 |
78 | 55, 74, 77 | 3jaodan 1295 | . . . . 5 |
79 | 2, 78 | sylan2b 285 | . . . 4 |
80 | 32, 38, 79 | 3jaoian 1294 | . . 3 |
81 | 1, 80 | sylanb 282 | . 2 |
82 | 81 | 3impia 1189 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 w3o 966 w3a 967 wceq 1342 wcel 2135 wrex 2443 class class class wbr 3976 (class class class)co 5836 cr 7743 c1 7745 caddc 7747 cpnf 7921 cmnf 7922 cxr 7923 clt 7924 cmin 8060 cz 9182 cq 9548 |
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-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 ax-pre-mulext 7862 ax-arch 7863 |
This theorem depends on definitions: df-bi 116 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-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 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-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-id 4265 df-po 4268 df-iso 4269 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-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-reap 8464 df-ap 8471 df-div 8560 df-inn 8849 df-2 8907 df-n0 9106 df-z 9183 df-uz 9458 df-q 9549 df-rp 9581 |
This theorem is referenced by: ioo0 10185 ioom 10186 ico0 10187 ioc0 10188 blssps 12968 blss 12969 tgqioo 13088 |
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