Nearby theorems
|
|
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
| Mirrors > Home > ILE Home > Th. List > apcotr | Unicode version | ||
| Description: Apartness is cotransitive. (Contributed by Jim Kingdon, 16-Feb-2020.) |
| Ref | Expression |
|---|---|
| apcotr |
     ![/\](wedge.gif "/\"){kind=small} 
   #
#  # |
are mutually distinct and distinct from all other
variables.| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cnre 7905 |
. . 3
      | |
| 2 | 1 | 3ad2ant3 1015 |
. 2
|
| 3 | cnre 7905 |
. . . . . . 7
| |
| 4 | 3 | 3ad2ant2 1014 |
. . . . . 6
|
| 5 | 4 | ad2antrr 485 |
. . . . 5
|
| 6 | cnre 7905 |
. . . . . . . . . . 11
| |
| 7 | 6 | 3ad2ant1 1013 |
. . . . . . . . . 10
|
| 8 | 7 | adantr 274 |
. . . . . . . . 9
|
| 9 | 8 | ad3antrrr 489 |
. . . . . . . 8
|
| 10 | simpr 109 |
. . . . . . . . . . . . . . 15
| |
| 11 | simpllr 529 |
. . . . . . . . . . . . . . 15
| |
| 12 | 10, 11 | breq12d 4000 |
. . . . . . . . . . . . . 14
#
# |
| 13 | simplrl 530 |
. . . . . . . . . . . . . . 15
| |
| 14 | simplrr 531 |
. . . . . . . . . . . . . . 15
| |
| 15 | simprl 526 |
. . . . . . . . . . . . . . . 16
| |
| 16 | 15 | ad3antrrr 489 |
. . . . . . . . . . . . . . 15
|
| 17 | simprr 527 |
. . . . . . . . . . . . . . . 16
| |
| 18 | 17 | ad3antrrr 489 |
. . . . . . . . . . . . . . 15
|
| 19 | apreim 8511 |
. . . . . . . . . . . . . . 15
#
#
# | |
| 20 | 13, 14, 16, 18, 19 | syl22anc 1234 |
. . . . . . . . . . . . . 14
#
#
# |
| 21 | 12, 20 | bitrd 187 |
. . . . . . . . . . . . 13
#
# # |
| 22 | simprl 526 |
. . . . . . . . . . . . . . . . 17
| |
| 23 | 22 | ad2antrr 485 |
. . . . . . . . . . . . . . . 16
|
| 24 | 23 | ad3antrrr 489 |
. . . . . . . . . . . . . . 15
|
| 25 | reapcotr 8506 |
. . . . . . . . . . . . . . 15
# # # | |
| 26 | 13, 16, 24, 25 | syl3anc 1233 |
. . . . . . . . . . . . . 14
# # # |
| 27 | simprr 527 |
. . . . . . . . . . . . . . . . 17
| |
| 28 | 27 | ad2antrr 485 |
. . . . . . . . . . . . . . . 16
|
| 29 | 28 | ad3antrrr 489 |
. . . . . . . . . . . . . . 15
|
| 30 | reapcotr 8506 |
. . . . . . . . . . . . . . 15
# # # | |
| 31 | 14, 18, 29, 30 | syl3anc 1233 |
. . . . . . . . . . . . . 14
# # # |
| 32 | 26, 31 | orim12d 781 |
. . . . . . . . . . . . 13
# # # #
# # |
| 33 | 21, 32 | sylbid 149 |
. . . . . . . . . . . 12
#
# # # # |
| 34 | or4 766 |
. . . . . . . . . . . 12
# # # # # #
# # | |
| 35 | 33, 34 | syl6ib 160 |
. . . . . . . . . . 11
#
# # # # |
| 36 | simplr 525 |
. . . . . . . . . . . . . . 15
| |
| 37 | 36 | ad3antrrr 489 |
. . . . . . . . . . . . . 14
|
| 38 | 10, 37 | breq12d 4000 |
. . . . . . . . . . . . 13
#
# |
| 39 | apreim 8511 |
. . . . . . . . . . . . . 14
#
#
# | |
| 40 | 13, 14, 24, 29, 39 | syl22anc 1234 |
. . . . . . . . . . . . 13
#
#
# |
| 41 | 38, 40 | bitrd 187 |
. . . . . . . . . . . 12
#
# #
|
| 42 | 11, 37 | breq12d 4000 |
. . . . . . . . . . . . 13
#
# |
| 43 | apreim 8511 |
. . . . . . . . . . . . . 14
#
#
# | |
| 44 | 16, 18, 24, 29, 43 | syl22anc 1234 |
. . . . . . . . . . . . 13
#
#
# |
| 45 | 42, 44 | bitrd 187 |
. . . . . . . . . . . 12
#
# # |
| 46 | 41, 45 | orbi12d 788 |
. . . . . . . . . . 11
# #
# # # # |
| 47 | 35, 46 | sylibrd 168 |
. . . . . . . . . 10
#
# # |
| 48 | 47 | ex 114 |
. . . . . . . . 9
#
# # |
| 49 | 48 | rexlimdvva 2595 |
. . . . . . . 8
#
# # |
| 50 | 9, 49 | mpd 13 |
. . . . . . 7
#
# # |
| 51 | 50 | ex 114 |
. . . . . 6
#
# # |
| 52 | 51 | rexlimdvva 2595 |
. . . . 5
#
# # |
| 53 | 5, 52 | mpd 13 |
. . . 4
#
# # |
| 54 | 53 | ex 114 |
. . 3
#
# # |
| 55 | 54 | rexlimdvva 2595 |
. 2
#
# # |
| 56 | 2, 55 | mpd 13 |
1
#
# # |
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 103
wb 104 wo 703 w3a 973 wceq 1348 wcel 2141 wrex 2449 class class class wbr 3987
(class class class)co 5851 cc 7761 cr 7762 ci 7765 caddc 7766 cmul 7768 # cap 8489 |
| 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 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-cnex 7854 ax-resscn 7855 ax-1cn 7856 ax-1re 7857 ax-icn 7858 ax-addcl 7859 ax-addrcl 7860 ax-mulcl 7861 ax-mulrcl 7862 ax-addcom 7863 ax-mulcom 7864 ax-addass 7865 ax-mulass 7866 ax-distr 7867 ax-i2m1 7868 ax-0lt1 7869 ax-1rid 7870 ax-0id 7871 ax-rnegex 7872 ax-precex 7873 ax-cnre 7874 ax-pre-ltirr 7875 ax-pre-ltwlin 7876 ax-pre-lttrn 7877 ax-pre-apti 7878 ax-pre-ltadd 7879 ax-pre-mulgt0 7880 |
| 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-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-iota 5158 df-fun 5198 df-fv 5204 df-riota 5807 df-ov 5854 df-oprab 5855 df-mpo 5856 df-pnf 7945 df-mnf 7946 df-ltxr 7948 df-sub 8081 df-neg 8082 df-reap 8483 df-ap 8490 |
| This theorem is referenced by: addext 8518 mulext 8522 mul0eqap 8577 |
| Copyright terms: Public domain | W3C validator |