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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 8138 |
. . 3
      | |
| 2 | 1 | 3ad2ant3 1044 |
. 2
|
| 3 | cnre 8138 |
. . . . . . 7
| |
| 4 | 3 | 3ad2ant2 1043 |
. . . . . 6
|
| 5 | 4 | ad2antrr 488 |
. . . . 5
|
| 6 | cnre 8138 |
. . . . . . . . . . 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 4095 |
. . . . . . . . . . . . . 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 8746 |
. . . . . . . . . . . . . . 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 8741 |
. . . . . . . . . . . . . . 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 8741 |
. . . . . . . . . . . . . . 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 4095 |
. . . . . . . . . . . . 13
#
# |
| 39 | apreim 8746 |
. . . . . . . . . . . . . 14
#
#
# | |
| 40 | 13, 14, 24, 29, 39 | syl22anc 1272 |
. . . . . . . . . . . . 13
#
#
# |
| 41 | 38, 40 | bitrd 188 |
. . . . . . . . . . . 12
#
# #
|
| 42 | 11, 37 | breq12d 4095 |
. . . . . . . . . . . . 13
#
# |
| 43 | apreim 8746 |
. . . . . . . . . . . . . 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 4082
(class class class)co 6000 cc 7993 cr 7994 ci 7997 caddc 7998 cmul 8000 # cap 8724 |
| 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 4201 ax-pow 4257 ax-pr 4292 ax-un 4523 ax-setind 4628 ax-cnex 8086 ax-resscn 8087 ax-1cn 8088 ax-1re 8089 ax-icn 8090 ax-addcl 8091 ax-addrcl 8092 ax-mulcl 8093 ax-mulrcl 8094 ax-addcom 8095 ax-mulcom 8096 ax-addass 8097 ax-mulass 8098 ax-distr 8099 ax-i2m1 8100 ax-0lt1 8101 ax-1rid 8102 ax-0id 8103 ax-rnegex 8104 ax-precex 8105 ax-cnre 8106 ax-pre-ltirr 8107 ax-pre-ltwlin 8108 ax-pre-lttrn 8109 ax-pre-apti 8110 ax-pre-ltadd 8111 ax-pre-mulgt0 8112 |
| 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 2801 df-sbc 3029 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3888 df-br 4083 df-opab 4145 df-id 4383 df-xp 4724 df-rel 4725 df-cnv 4726 df-co 4727 df-dm 4728 df-iota 5277 df-fun 5319 df-fv 5325 df-riota 5953 df-ov 6003 df-oprab 6004 df-mpo 6005 df-pnf 8179 df-mnf 8180 df-ltxr 8182 df-sub 8315 df-neg 8316 df-reap 8718 df-ap 8725 |
| This theorem is referenced by: addext 8753 mulext 8757 aptap 8793 mul0eqap 8813 |
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