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Mirrors > Home > ILE Home > Th. List > reaplt | Unicode version |
Description: Real apartness in terms of less than. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by Jim Kingdon, 1-Feb-2020.) |
Ref | Expression |
---|---|
reaplt | # |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | apreap 7687 | . 2 # #ℝ | |
2 | reapval 7676 | . 2 #ℝ | |
3 | 1, 2 | bitrd 186 | 1 # |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wo 661 wcel 1433 class class class wbr 3785 cr 6980 clt 7153 #ℝ creap 7674 # cap 7681 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-mulrcl 7075 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-precex 7086 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-lttrn 7090 ax-pre-apti 7091 ax-pre-ltadd 7092 ax-pre-mulgt0 7093 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-pnf 7155 df-mnf 7156 df-ltxr 7158 df-sub 7281 df-neg 7282 df-reap 7675 df-ap 7682 |
This theorem is referenced by: reapltxor 7689 1ap0 7690 reapmul1lem 7694 reapmul1 7695 reapadd1 7696 reapneg 7697 reapcotr 7698 remulext1 7699 apsqgt0 7701 apsym 7706 msqge0 7716 mulge0 7719 leltap 7724 gt0ap0 7725 ltleap 7730 ltap 7731 ap0gt0 7738 recexaplem2 7742 zapne 8422 qlttri2 8726 sq11ap 9639 nn0opthd 9649 recvguniq 9881 sqrt11ap 9924 ltabs 9973 |
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