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Mirrors > Home > ILE Home > Th. List > lttri3 | Unicode version |
Description: Tightness of real apartness. (Contributed by NM, 5-May-1999.) |
Ref | Expression |
---|---|
lttri3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltnr 7834 | . . . . 5 | |
2 | breq2 3928 | . . . . . 6 | |
3 | 2 | notbid 656 | . . . . 5 |
4 | 1, 3 | syl5ibcom 154 | . . . 4 |
5 | breq1 3927 | . . . . . 6 | |
6 | 5 | notbid 656 | . . . . 5 |
7 | 1, 6 | syl5ibcom 154 | . . . 4 |
8 | 4, 7 | jcad 305 | . . 3 |
9 | 8 | adantr 274 | . 2 |
10 | ioran 741 | . . 3 | |
11 | axapti 7828 | . . . 4 | |
12 | 11 | 3expia 1183 | . . 3 |
13 | 10, 12 | syl5bir 152 | . 2 |
14 | 9, 13 | impbid 128 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 697 wceq 1331 wcel 1480 class class class wbr 3924 cr 7612 clt 7793 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-pre-ltirr 7725 ax-pre-apti 7728 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-xp 4540 df-pnf 7795 df-mnf 7796 df-ltxr 7798 |
This theorem is referenced by: letri3 7838 lttri3i 7854 lttri3d 7871 inelr 8339 lbinf 8699 suprubex 8702 suprlubex 8703 suprleubex 8705 sup3exmid 8708 suprzclex 9142 infrenegsupex 9382 supminfex 9385 xrlttri3 9576 maxleim 10970 maxabs 10974 maxleast 10978 zsupcl 11629 zssinfcl 11630 infssuzledc 11632 dvdslegcd 11642 bezoutlemsup 11686 dfgcd2 11691 lcmgcdlem 11747 suplociccex 12761 pilem3 12853 taupi 13228 |
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