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Mirrors > Home > ILE Home > Th. List > archpr | Unicode version |
Description: For any positive real, there is an integer that is greater than it. This is also known as the "archimedean property". The integer is embedded into the reals as described at nnprlu 7485. (Contributed by Jim Kingdon, 22-Apr-2020.) |
Ref | Expression |
---|---|
archpr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prop 7407 | . . 3 | |
2 | prmu 7410 | . . 3 | |
3 | 1, 2 | syl 14 | . 2 |
4 | archnqq 7349 | . . . 4 | |
5 | 4 | ad2antrl 482 | . . 3 |
6 | simprl 521 | . . . . . . . 8 | |
7 | 6 | ad2antrr 480 | . . . . . . 7 |
8 | simprr 522 | . . . . . . . 8 | |
9 | 8 | ad2antrr 480 | . . . . . . 7 |
10 | simpr 109 | . . . . . . . 8 | |
11 | vex 2724 | . . . . . . . . 9 | |
12 | breq1 3979 | . . . . . . . . 9 | |
13 | ltnqex 7481 | . . . . . . . . . 10 | |
14 | gtnqex 7482 | . . . . . . . . . 10 | |
15 | 13, 14 | op1st 6106 | . . . . . . . . 9 |
16 | 11, 12, 15 | elab2 2869 | . . . . . . . 8 |
17 | 10, 16 | sylibr 133 | . . . . . . 7 |
18 | eleq1 2227 | . . . . . . . . 9 | |
19 | eleq1 2227 | . . . . . . . . 9 | |
20 | 18, 19 | anbi12d 465 | . . . . . . . 8 |
21 | 20 | rspcev 2825 | . . . . . . 7 |
22 | 7, 9, 17, 21 | syl12anc 1225 | . . . . . 6 |
23 | simplll 523 | . . . . . . 7 | |
24 | nnprlu 7485 | . . . . . . . 8 | |
25 | 24 | ad2antlr 481 | . . . . . . 7 |
26 | ltdfpr 7438 | . . . . . . 7 | |
27 | 23, 25, 26 | syl2anc 409 | . . . . . 6 |
28 | 22, 27 | mpbird 166 | . . . . 5 |
29 | 28 | ex 114 | . . . 4 |
30 | 29 | reximdva 2566 | . . 3 |
31 | 5, 30 | mpd 13 | . 2 |
32 | 3, 31 | rexlimddv 2586 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wcel 2135 cab 2150 wrex 2443 cop 3573 class class class wbr 3976 cfv 5182 c1st 6098 c2nd 6099 c1o 6368 cec 6490 cnpi 7204 ceq 7211 cnq 7212 cltq 7217 cnp 7223 cltp 7227 |
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 df-inp 7398 df-iltp 7402 |
This theorem is referenced by: archsr 7714 |
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