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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | apreap 8373 |
. 2
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2 | reapval 8362 |
. 2
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3 | 1, 2 | bitrd 187 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-mulrcl 7743 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-precex 7754 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 ax-pre-mulgt0 7761 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-iota 5096 df-fun 5133 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-pnf 7826 df-mnf 7827 df-ltxr 7829 df-sub 7959 df-neg 7960 df-reap 8361 df-ap 8368 |
This theorem is referenced by: reapltxor 8375 1ap0 8376 reapmul1lem 8380 reapmul1 8381 reapadd1 8382 reapneg 8383 reapcotr 8384 remulext1 8385 apsqgt0 8387 apsym 8392 msqge0 8402 mulge0 8405 leltap 8411 gt0ap0 8412 ltleap 8418 ltap 8419 ap0gt0 8426 recexaplem2 8437 zapne 9149 qlttri2 9460 apbtwnz 10078 sq11ap 10489 nn0opthd 10500 recvguniq 10799 sqrt11ap 10842 ltabs 10891 reopnap 12746 dedekindeu 12809 dedekindicclemicc 12818 ivthinc 12829 reapef 12907 coseq0q4123 12963 cos11 12982 logrpap0b 13005 triap 13399 trirec0 13412 apdifflemf 13414 neapmkvlem 13424 |
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