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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 8165 |
. . 3
      | |
| 2 | 1 | 3ad2ant3 1044 |
. 2
|
| 3 | cnre 8165 |
. . . . . . 7
| |
| 4 | 3 | 3ad2ant2 1043 |
. . . . . 6
|
| 5 | 4 | ad2antrr 488 |
. . . . 5
|
| 6 | cnre 8165 |
. . . . . . . . . . 11
| |
| 7 | 6 | 3ad2ant1 1042 |
. . . . . . . . . 10
|
| 8 | 7 | adantr 276 |
. . . . . . . . 9
|
| 9 | 8 | ad3antrrr 492 |
. . . . . . . 8
|
| 10 | simpr 110 |
. . . . . . . . . . . . . . 15
| |
| 11 | simpllr 534 |
. . . . . . . . . . . . . . 15
| |
| 12 | 10, 11 | breq12d 4099 |
. . . . . . . . . . . . . 14
#
# |
| 13 | simplrl 535 |
. . . . . . . . . . . . . . 15
| |
| 14 | simplrr 536 |
. . . . . . . . . . . . . . 15
| |
| 15 | simprl 529 |
. . . . . . . . . . . . . . . 16
| |
| 16 | 15 | ad3antrrr 492 |
. . . . . . . . . . . . . . 15
|
| 17 | simprr 531 |
. . . . . . . . . . . . . . . 16
| |
| 18 | 17 | ad3antrrr 492 |
. . . . . . . . . . . . . . 15
|
| 19 | apreim 8773 |
. . . . . . . . . . . . . . 15
#
#
# | |
| 20 | 13, 14, 16, 18, 19 | syl22anc 1272 |
. . . . . . . . . . . . . 14
#
#
# |
| 21 | 12, 20 | bitrd 188 |
. . . . . . . . . . . . 13
#
# # |
| 22 | simprl 529 |
. . . . . . . . . . . . . . . . 17
| |
| 23 | 22 | ad2antrr 488 |
. . . . . . . . . . . . . . . 16
|
| 24 | 23 | ad3antrrr 492 |
. . . . . . . . . . . . . . 15
|
| 25 | reapcotr 8768 |
. . . . . . . . . . . . . . 15
# # # | |
| 26 | 13, 16, 24, 25 | syl3anc 1271 |
. . . . . . . . . . . . . 14
# # # |
| 27 | simprr 531 |
. . . . . . . . . . . . . . . . 17
| |
| 28 | 27 | ad2antrr 488 |
. . . . . . . . . . . . . . . 16
|
| 29 | 28 | ad3antrrr 492 |
. . . . . . . . . . . . . . 15
|
| 30 | reapcotr 8768 |
. . . . . . . . . . . . . . 15
# # # | |
| 31 | 14, 18, 29, 30 | syl3anc 1271 |
. . . . . . . . . . . . . 14
# # # |
| 32 | 26, 31 | orim12d 791 |
. . . . . . . . . . . . 13
# # # #
# # |
| 33 | 21, 32 | sylbid 150 |
. . . . . . . . . . . 12
#
# # # # |
| 34 | or4 776 |
. . . . . . . . . . . 12
# # # # # #
# # | |
| 35 | 33, 34 | imbitrdi 161 |
. . . . . . . . . . 11
#
# # # # |
| 36 | simplr 528 |
. . . . . . . . . . . . . . 15
| |
| 37 | 36 | ad3antrrr 492 |
. . . . . . . . . . . . . 14
|
| 38 | 10, 37 | breq12d 4099 |
. . . . . . . . . . . . 13
#
# |
| 39 | apreim 8773 |
. . . . . . . . . . . . . 14
#
#
# | |
| 40 | 13, 14, 24, 29, 39 | syl22anc 1272 |
. . . . . . . . . . . . 13
#
#
# |
| 41 | 38, 40 | bitrd 188 |
. . . . . . . . . . . 12
#
# #
|
| 42 | 11, 37 | breq12d 4099 |
. . . . . . . . . . . . 13
#
# |
| 43 | apreim 8773 |
. . . . . . . . . . . . . 14
#
#
# | |
| 44 | 16, 18, 24, 29, 43 | syl22anc 1272 |
. . . . . . . . . . . . 13
#
#
# |
| 45 | 42, 44 | bitrd 188 |
. . . . . . . . . . . 12
#
# # |
| 46 | 41, 45 | orbi12d 798 |
. . . . . . . . . . 11
# #
# # # # |
| 47 | 35, 46 | sylibrd 169 |
. . . . . . . . . 10
#
# # |
| 48 | 47 | ex 115 |
. . . . . . . . 9
#
# # |
| 49 | 48 | rexlimdvva 2656 |
. . . . . . . 8
#
# # |
| 50 | 9, 49 | mpd 13 |
. . . . . . 7
#
# # |
| 51 | 50 | ex 115 |
. . . . . 6
#
# # |
| 52 | 51 | rexlimdvva 2656 |
. . . . 5
#
# # |
| 53 | 5, 52 | mpd 13 |
. . . 4
#
# # |
| 54 | 53 | ex 115 |
. . 3
#
# # |
| 55 | 54 | rexlimdvva 2656 |
. 2
#
# # |
| 56 | 2, 55 | mpd 13 |
1
#
# # |
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 104
wb 105 wo 713 w3a 1002 wceq 1395 wcel 2200 wrex 2509 class class class wbr 4086
(class class class)co 6013 cc 8020 cr 8021 ci 8024 caddc 8025 cmul 8027 # cap 8751 |
| 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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-sep 4205 ax-pow 4262 ax-pr 4297 ax-un 4528 ax-setind 4633 ax-cnex 8113 ax-resscn 8114 ax-1cn 8115 ax-1re 8116 ax-icn 8117 ax-addcl 8118 ax-addrcl 8119 ax-mulcl 8120 ax-mulrcl 8121 ax-addcom 8122 ax-mulcom 8123 ax-addass 8124 ax-mulass 8125 ax-distr 8126 ax-i2m1 8127 ax-0lt1 8128 ax-1rid 8129 ax-0id 8130 ax-rnegex 8131 ax-precex 8132 ax-cnre 8133 ax-pre-ltirr 8134 ax-pre-ltwlin 8135 ax-pre-lttrn 8136 ax-pre-apti 8137 ax-pre-ltadd 8138 ax-pre-mulgt0 8139 |
| This theorem depends on definitions: df-bi 117 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rab 2517 df-v 2802 df-sbc 3030 df-dif 3200 df-un 3202 df-in 3204 df-ss 3211 df-pw 3652 df-sn 3673 df-pr 3674 df-op 3676 df-uni 3892 df-br 4087 df-opab 4149 df-id 4388 df-xp 4729 df-rel 4730 df-cnv 4731 df-co 4732 df-dm 4733 df-iota 5284 df-fun 5326 df-fv 5332 df-riota 5966 df-ov 6016 df-oprab 6017 df-mpo 6018 df-pnf 8206 df-mnf 8207 df-ltxr 8209 df-sub 8342 df-neg 8343 df-reap 8745 df-ap 8752 |
| This theorem is referenced by: addext 8780 mulext 8784 aptap 8820 mul0eqap 8840 |
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