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Mirrors > Home > MPE Home > Th. List > nqpr | Structured version Visualization version Unicode version |
Description: The canonical embedding of the rationals into the reals. (Contributed by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nqpr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nsmallnq 9799 | . . . . 5 | |
2 | abn0 3954 | . . . . 5 | |
3 | 1, 2 | sylibr 224 | . . . 4 |
4 | 0pss 4013 | . . . 4 | |
5 | 3, 4 | sylibr 224 | . . 3 |
6 | ltrelnq 9748 | . . . . . . 7 | |
7 | 6 | brel 5168 | . . . . . 6 |
8 | 7 | simpld 475 | . . . . 5 |
9 | 8 | abssi 3677 | . . . 4 |
10 | ltsonq 9791 | . . . . . . 7 | |
11 | 10, 6 | soirri 5522 | . . . . . 6 |
12 | breq1 4656 | . . . . . . 7 | |
13 | 12 | elabg 3351 | . . . . . 6 |
14 | 11, 13 | mtbiri 317 | . . . . 5 |
15 | 14 | ancli 574 | . . . 4 |
16 | ssnelpss 3718 | . . . 4 | |
17 | 9, 15, 16 | mpsyl 68 | . . 3 |
18 | 5, 17 | jca 554 | . 2 |
19 | vex 3203 | . . . . 5 | |
20 | breq1 4656 | . . . . 5 | |
21 | 19, 20 | elab 3350 | . . . 4 |
22 | 10, 6 | sotri 5523 | . . . . . . . . 9 |
23 | 22 | expcom 451 | . . . . . . . 8 |
24 | 23 | adantl 482 | . . . . . . 7 |
25 | vex 3203 | . . . . . . . 8 | |
26 | breq1 4656 | . . . . . . . 8 | |
27 | 25, 26 | elab 3350 | . . . . . . 7 |
28 | 24, 27 | syl6ibr 242 | . . . . . 6 |
29 | 28 | alrimiv 1855 | . . . . 5 |
30 | ltbtwnnq 9800 | . . . . . . . 8 | |
31 | 27 | anbi2i 730 | . . . . . . . . . . 11 |
32 | 31 | biimpri 218 | . . . . . . . . . 10 |
33 | 32 | ancomd 467 | . . . . . . . . 9 |
34 | 33 | eximi 1762 | . . . . . . . 8 |
35 | 30, 34 | sylbi 207 | . . . . . . 7 |
36 | 35 | adantl 482 | . . . . . 6 |
37 | df-rex 2918 | . . . . . 6 | |
38 | 36, 37 | sylibr 224 | . . . . 5 |
39 | 29, 38 | jca 554 | . . . 4 |
40 | 21, 39 | sylan2b 492 | . . 3 |
41 | 40 | ralrimiva 2966 | . 2 |
42 | elnp 9809 | . 2 | |
43 | 18, 41, 42 | sylanbrc 698 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wa 384 wal 1481 wex 1704 wcel 1990 cab 2608 wne 2794 wral 2912 wrex 2913 wss 3574 wpss 3575 c0 3915 class class class wbr 4653 cnq 9674 cltq 9680 cnp 9681 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-omul 7565 df-er 7742 df-ni 9694 df-pli 9695 df-mi 9696 df-lti 9697 df-plpq 9730 df-mpq 9731 df-ltpq 9732 df-enq 9733 df-nq 9734 df-erq 9735 df-plq 9736 df-mq 9737 df-1nq 9738 df-rq 9739 df-ltnq 9740 df-np 9803 |
This theorem is referenced by: 1pr 9837 |
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