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Mirrors > Home > ILE Home > Th. List > dfrp2 | Unicode version |
Description: Alternate definition of the positive real numbers. (Contributed by Thierry Arnoux, 4-May-2020.) |
Ref | Expression |
---|---|
dfrp2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltpnf 9682 | . . . . . 6 | |
2 | 1 | adantr 274 | . . . . 5 |
3 | 2 | pm4.71i 389 | . . . 4 |
4 | df-3an 965 | . . . 4 | |
5 | 3, 4 | bitr4i 186 | . . 3 |
6 | elrp 9557 | . . 3 | |
7 | 0xr 7919 | . . . 4 | |
8 | pnfxr 7925 | . . . 4 | |
9 | elioo2 9820 | . . . 4 | |
10 | 7, 8, 9 | mp2an 423 | . . 3 |
11 | 5, 6, 10 | 3bitr4i 211 | . 2 |
12 | 11 | eqriv 2154 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 w3a 963 wceq 1335 wcel 2128 class class class wbr 3965 (class class class)co 5821 cr 7726 cc0 7727 cpnf 7904 cxr 7906 clt 7907 crp 9555 cioo 9787 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4495 ax-cnex 7818 ax-resscn 7819 ax-1re 7821 ax-addrcl 7824 ax-rnegex 7836 ax-pre-ltirr 7839 ax-pre-ltwlin 7840 ax-pre-lttrn 7841 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-br 3966 df-opab 4026 df-id 4253 df-po 4256 df-iso 4257 df-xp 4591 df-rel 4592 df-cnv 4593 df-co 4594 df-dm 4595 df-iota 5134 df-fun 5171 df-fv 5177 df-ov 5824 df-oprab 5825 df-mpo 5826 df-pnf 7909 df-mnf 7910 df-xr 7911 df-ltxr 7912 df-le 7913 df-rp 9556 df-ioo 9791 |
This theorem is referenced by: (None) |
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