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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltnr 8064 |
. . . . 5
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2 | breq2 4022 |
. . . . . 6
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3 | 2 | notbid 668 |
. . . . 5
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4 | 1, 3 | syl5ibcom 155 |
. . . 4
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5 | breq1 4021 |
. . . . . 6
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6 | 5 | notbid 668 |
. . . . 5
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7 | 1, 6 | syl5ibcom 155 |
. . . 4
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8 | 4, 7 | jcad 307 |
. . 3
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9 | 8 | adantr 276 |
. 2
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10 | ioran 753 |
. . 3
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11 | axapti 8058 |
. . . 4
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12 | 11 | 3expia 1207 |
. . 3
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13 | 10, 12 | biimtrrid 153 |
. 2
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14 | 9, 13 | impbid 129 |
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 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7932 ax-resscn 7933 ax-pre-ltirr 7953 ax-pre-apti 7956 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-xp 4650 df-pnf 8024 df-mnf 8025 df-ltxr 8027 |
This theorem is referenced by: letri3 8068 lttri3i 8085 lttri3d 8102 inelr 8571 lbinf 8935 suprubex 8938 suprlubex 8939 suprleubex 8941 sup3exmid 8944 suprzclex 9381 infrenegsupex 9624 supminfex 9627 infregelbex 9628 xrlttri3 9827 maxleim 11246 maxabs 11250 maxleast 11254 zsupcl 11980 zssinfcl 11981 infssuzledc 11983 suprzcl2dc 11988 dvdslegcd 11997 bezoutlemsup 12042 dfgcd2 12047 lcmgcdlem 12109 suplociccex 14560 pilem3 14661 taupi 15280 |
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