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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 9556 | . . 3 | |
2 | elxr 9556 | . . . . 5 | |
3 | qbtwnre 10027 | . . . . . . 7 | |
4 | 3 | 3expia 1183 | . . . . . 6 |
5 | simpl 108 | . . . . . . . . 9 | |
6 | peano2re 7891 | . . . . . . . . . 10 | |
7 | 6 | adantr 274 | . . . . . . . . 9 |
8 | ltp1 8595 | . . . . . . . . . 10 | |
9 | 8 | adantr 274 | . . . . . . . . 9 |
10 | qbtwnre 10027 | . . . . . . . . 9 | |
11 | 5, 7, 9, 10 | syl3anc 1216 | . . . . . . . 8 |
12 | qre 9410 | . . . . . . . . . . . . . 14 | |
13 | ltpnf 9560 | . . . . . . . . . . . . . 14 | |
14 | 12, 13 | syl 14 | . . . . . . . . . . . . 13 |
15 | 14 | adantl 275 | . . . . . . . . . . . 12 |
16 | simplr 519 | . . . . . . . . . . . 12 | |
17 | 15, 16 | breqtrrd 3951 | . . . . . . . . . . 11 |
18 | 17 | a1d 22 | . . . . . . . . . 10 |
19 | 18 | anim2d 335 | . . . . . . . . 9 |
20 | 19 | reximdva 2532 | . . . . . . . 8 |
21 | 11, 20 | mpd 13 | . . . . . . 7 |
22 | 21 | a1d 22 | . . . . . 6 |
23 | rexr 7804 | . . . . . . 7 | |
24 | breq2 3928 | . . . . . . . . 9 | |
25 | 24 | adantl 275 | . . . . . . . 8 |
26 | nltmnf 9567 | . . . . . . . . . 10 | |
27 | 26 | adantr 274 | . . . . . . . . 9 |
28 | 27 | pm2.21d 608 | . . . . . . . 8 |
29 | 25, 28 | sylbid 149 | . . . . . . 7 |
30 | 23, 29 | sylan 281 | . . . . . 6 |
31 | 4, 22, 30 | 3jaodan 1284 | . . . . 5 |
32 | 2, 31 | sylan2b 285 | . . . 4 |
33 | breq1 3927 | . . . . . 6 | |
34 | 33 | adantr 274 | . . . . 5 |
35 | pnfnlt 9566 | . . . . . . 7 | |
36 | 35 | adantl 275 | . . . . . 6 |
37 | 36 | pm2.21d 608 | . . . . 5 |
38 | 34, 37 | sylbid 149 | . . . 4 |
39 | peano2rem 8022 | . . . . . . . . . 10 | |
40 | 39 | adantl 275 | . . . . . . . . 9 |
41 | simpr 109 | . . . . . . . . 9 | |
42 | ltm1 8597 | . . . . . . . . . 10 | |
43 | 42 | adantl 275 | . . . . . . . . 9 |
44 | qbtwnre 10027 | . . . . . . . . 9 | |
45 | 40, 41, 43, 44 | syl3anc 1216 | . . . . . . . 8 |
46 | simpll 518 | . . . . . . . . . . . 12 | |
47 | 12 | adantl 275 | . . . . . . . . . . . . 13 |
48 | mnflt 9562 | . . . . . . . . . . . . 13 | |
49 | 47, 48 | syl 14 | . . . . . . . . . . . 12 |
50 | 46, 49 | eqbrtrd 3945 | . . . . . . . . . . 11 |
51 | 50 | a1d 22 | . . . . . . . . . 10 |
52 | 51 | anim1d 334 | . . . . . . . . 9 |
53 | 52 | reximdva 2532 | . . . . . . . 8 |
54 | 45, 53 | mpd 13 | . . . . . . 7 |
55 | 54 | a1d 22 | . . . . . 6 |
56 | 1re 7758 | . . . . . . . . . 10 | |
57 | mnflt 9562 | . . . . . . . . . 10 | |
58 | 56, 57 | ax-mp 5 | . . . . . . . . 9 |
59 | breq1 3927 | . . . . . . . . 9 | |
60 | 58, 59 | mpbiri 167 | . . . . . . . 8 |
61 | ltpnf 9560 | . . . . . . . . . 10 | |
62 | 56, 61 | ax-mp 5 | . . . . . . . . 9 |
63 | breq2 3928 | . . . . . . . . 9 | |
64 | 62, 63 | mpbiri 167 | . . . . . . . 8 |
65 | 1z 9073 | . . . . . . . . . 10 | |
66 | zq 9411 | . . . . . . . . . 10 | |
67 | 65, 66 | ax-mp 5 | . . . . . . . . 9 |
68 | breq2 3928 | . . . . . . . . . . 11 | |
69 | breq1 3927 | . . . . . . . . . . 11 | |
70 | 68, 69 | anbi12d 464 | . . . . . . . . . 10 |
71 | 70 | rspcev 2784 | . . . . . . . . 9 |
72 | 67, 71 | mpan 420 | . . . . . . . 8 |
73 | 60, 64, 72 | syl2an 287 | . . . . . . 7 |
74 | 73 | a1d 22 | . . . . . 6 |
75 | 3mix3 1152 | . . . . . . . 8 | |
76 | 75, 1 | sylibr 133 | . . . . . . 7 |
77 | 76, 29 | sylan 281 | . . . . . 6 |
78 | 55, 74, 77 | 3jaodan 1284 | . . . . 5 |
79 | 2, 78 | sylan2b 285 | . . . 4 |
80 | 32, 38, 79 | 3jaoian 1283 | . . 3 |
81 | 1, 80 | sylanb 282 | . 2 |
82 | 81 | 3impia 1178 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 w3o 961 w3a 962 wceq 1331 wcel 1480 wrex 2415 class class class wbr 3924 (class class class)co 5767 cr 7612 c1 7614 caddc 7616 cpnf 7790 cmnf 7791 cxr 7792 clt 7793 cmin 7926 cz 9047 cq 9404 |
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-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-mulrcl 7712 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-1rid 7720 ax-0id 7721 ax-rnegex 7722 ax-precex 7723 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-apti 7728 ax-pre-ltadd 7729 ax-pre-mulgt0 7730 ax-pre-mulext 7731 ax-arch 7732 |
This theorem depends on definitions: df-bi 116 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-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rmo 2422 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-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-id 4210 df-po 4213 df-iso 4214 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-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-reap 8330 df-ap 8337 df-div 8426 df-inn 8714 df-2 8772 df-n0 8971 df-z 9048 df-uz 9320 df-q 9405 df-rp 9435 |
This theorem is referenced by: ioo0 10030 ioom 10031 ico0 10032 ioc0 10033 blssps 12585 blss 12586 tgqioo 12705 |
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