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| 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 8270 |
. . 3
      | |
| 2 | 1 | 3ad2ant3 1047 |
. 2
|
| 3 | cnre 8270 |
. . . . . . 7
| |
| 4 | 3 | 3ad2ant2 1046 |
. . . . . 6
|
| 5 | 4 | ad2antrr 488 |
. . . . 5
|
| 6 | cnre 8270 |
. . . . . . . . . . 11
| |
| 7 | 6 | 3ad2ant1 1045 |
. . . . . . . . . 10
|
| 8 | 7 | adantr 276 |
. . . . . . . . 9
|
| 9 | 8 | ad3antrrr 492 |
. . . . . . . 8
|
| 10 | simpr 110 |
. . . . . . . . . . . . . . 15
| |
| 11 | simpllr 536 |
. . . . . . . . . . . . . . 15
| |
| 12 | 10, 11 | breq12d 4122 |
. . . . . . . . . . . . . 14
#
# |
| 13 | simplrl 537 |
. . . . . . . . . . . . . . 15
| |
| 14 | simplrr 538 |
. . . . . . . . . . . . . . 15
| |
| 15 | simprl 531 |
. . . . . . . . . . . . . . . 16
| |
| 16 | 15 | ad3antrrr 492 |
. . . . . . . . . . . . . . 15
|
| 17 | simprr 533 |
. . . . . . . . . . . . . . . 16
| |
| 18 | 17 | ad3antrrr 492 |
. . . . . . . . . . . . . . 15
|
| 19 | apreim 8877 |
. . . . . . . . . . . . . . 15
#
#
# | |
| 20 | 13, 14, 16, 18, 19 | syl22anc 1275 |
. . . . . . . . . . . . . 14
#
#
# |
| 21 | 12, 20 | bitrd 188 |
. . . . . . . . . . . . 13
#
# # |
| 22 | simprl 531 |
. . . . . . . . . . . . . . . . 17
| |
| 23 | 22 | ad2antrr 488 |
. . . . . . . . . . . . . . . 16
|
| 24 | 23 | ad3antrrr 492 |
. . . . . . . . . . . . . . 15
|
| 25 | reapcotr 8872 |
. . . . . . . . . . . . . . 15
# # # | |
| 26 | 13, 16, 24, 25 | syl3anc 1274 |
. . . . . . . . . . . . . 14
# # # |
| 27 | simprr 533 |
. . . . . . . . . . . . . . . . 17
| |
| 28 | 27 | ad2antrr 488 |
. . . . . . . . . . . . . . . 16
|
| 29 | 28 | ad3antrrr 492 |
. . . . . . . . . . . . . . 15
|
| 30 | reapcotr 8872 |
. . . . . . . . . . . . . . 15
# # # | |
| 31 | 14, 18, 29, 30 | syl3anc 1274 |
. . . . . . . . . . . . . 14
# # # |
| 32 | 26, 31 | orim12d 794 |
. . . . . . . . . . . . 13
# # # #
# # |
| 33 | 21, 32 | sylbid 150 |
. . . . . . . . . . . 12
#
# # # # |
| 34 | or4 779 |
. . . . . . . . . . . 12
# # # # # #
# # | |
| 35 | 33, 34 | imbitrdi 161 |
. . . . . . . . . . 11
#
# # # # |
| 36 | simplr 529 |
. . . . . . . . . . . . . . 15
| |
| 37 | 36 | ad3antrrr 492 |
. . . . . . . . . . . . . 14
|
| 38 | 10, 37 | breq12d 4122 |
. . . . . . . . . . . . 13
#
# |
| 39 | apreim 8877 |
. . . . . . . . . . . . . 14
#
#
# | |
| 40 | 13, 14, 24, 29, 39 | syl22anc 1275 |
. . . . . . . . . . . . 13
#
#
# |
| 41 | 38, 40 | bitrd 188 |
. . . . . . . . . . . 12
#
# #
|
| 42 | 11, 37 | breq12d 4122 |
. . . . . . . . . . . . 13
#
# |
| 43 | apreim 8877 |
. . . . . . . . . . . . . 14
#
#
# | |
| 44 | 16, 18, 24, 29, 43 | syl22anc 1275 |
. . . . . . . . . . . . 13
#
#
# |
| 45 | 42, 44 | bitrd 188 |
. . . . . . . . . . . 12
#
# # |
| 46 | 41, 45 | orbi12d 801 |
. . . . . . . . . . 11
# #
# # # # |
| 47 | 35, 46 | sylibrd 169 |
. . . . . . . . . 10
#
# # |
| 48 | 47 | ex 115 |
. . . . . . . . 9
#
# # |
| 49 | 48 | rexlimdvva 2668 |
. . . . . . . 8
#
# # |
| 50 | 9, 49 | mpd 13 |
. . . . . . 7
#
# # |
| 51 | 50 | ex 115 |
. . . . . 6
#
# # |
| 52 | 51 | rexlimdvva 2668 |
. . . . 5
#
# # |
| 53 | 5, 52 | mpd 13 |
. . . 4
#
# # |
| 54 | 53 | ex 115 |
. . 3
#
# # |
| 55 | 54 | rexlimdvva 2668 |
. 2
#
# # |
| 56 | 2, 55 | mpd 13 |
1
#
# # |
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 104
wb 105 wo 716 w3a 1005 wceq 1398 wcel 2203 wrex 2521 class class class wbr 4109
(class class class)co 6050 cc 8125 cr 8126 ci 8129 caddc 8130 cmul 8132 # cap 8855 |
| 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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2205 ax-14 2206 ax-ext 2214 ax-sep 4228 ax-pow 4287 ax-pr 4322 ax-un 4554 ax-setind 4659 ax-cnex 8218 ax-resscn 8219 ax-1cn 8220 ax-1re 8221 ax-icn 8222 ax-addcl 8223 ax-addrcl 8224 ax-mulcl 8225 ax-mulrcl 8226 ax-addcom 8227 ax-mulcom 8228 ax-addass 8229 ax-mulass 8230 ax-distr 8231 ax-i2m1 8232 ax-0lt1 8233 ax-1rid 8234 ax-0id 8235 ax-rnegex 8236 ax-precex 8237 ax-cnre 8238 ax-pre-ltirr 8239 ax-pre-ltwlin 8240 ax-pre-lttrn 8241 ax-pre-apti 8242 ax-pre-ltadd 8243 ax-pre-mulgt0 8244 |
| This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-nel 2508 df-ral 2525 df-rex 2526 df-reu 2527 df-rab 2529 df-v 2815 df-sbc 3043 df-dif 3213 df-un 3215 df-in 3217 df-ss 3224 df-pw 3671 df-sn 3695 df-pr 3696 df-op 3698 df-uni 3915 df-br 4110 df-opab 4172 df-id 4414 df-xp 4755 df-rel 4756 df-cnv 4757 df-co 4758 df-dm 4759 df-iota 5312 df-fun 5354 df-fv 5360 df-riota 6003 df-ov 6053 df-oprab 6054 df-mpo 6055 df-pnf 8310 df-mnf 8311 df-ltxr 8313 df-sub 8446 df-neg 8447 df-reap 8849 df-ap 8856 |
| This theorem is referenced by: addext 8884 mulext 8888 aptap 8924 mul0eqap 8944 |
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