Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 7996 | . . . . 5 | |
2 | breq2 3993 | . . . . . 6 | |
3 | 2 | notbid 662 | . . . . 5 |
4 | 1, 3 | syl5ibcom 154 | . . . 4 |
5 | breq1 3992 | . . . . . 6 | |
6 | 5 | notbid 662 | . . . . 5 |
7 | 1, 6 | syl5ibcom 154 | . . . 4 |
8 | 4, 7 | jcad 305 | . . 3 |
9 | 8 | adantr 274 | . 2 |
10 | ioran 747 | . . 3 | |
11 | axapti 7990 | . . . 4 | |
12 | 11 | 3expia 1200 | . . 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 703 wceq 1348 wcel 2141 class class class wbr 3989 cr 7773 clt 7954 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-pre-ltirr 7886 ax-pre-apti 7889 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-xp 4617 df-pnf 7956 df-mnf 7957 df-ltxr 7959 |
This theorem is referenced by: letri3 8000 lttri3i 8017 lttri3d 8034 inelr 8503 lbinf 8864 suprubex 8867 suprlubex 8868 suprleubex 8870 sup3exmid 8873 suprzclex 9310 infrenegsupex 9553 supminfex 9556 infregelbex 9557 xrlttri3 9754 maxleim 11169 maxabs 11173 maxleast 11177 zsupcl 11902 zssinfcl 11903 infssuzledc 11905 suprzcl2dc 11910 dvdslegcd 11919 bezoutlemsup 11964 dfgcd2 11969 lcmgcdlem 12031 suplociccex 13397 pilem3 13498 taupi 14102 |
Copyright terms: Public domain | W3C validator |