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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 8088 |
. . 3
      | |
| 2 | 1 | 3ad2ant3 1023 |
. 2
|
| 3 | cnre 8088 |
. . . . . . 7
| |
| 4 | 3 | 3ad2ant2 1022 |
. . . . . 6
|
| 5 | 4 | ad2antrr 488 |
. . . . 5
|
| 6 | cnre 8088 |
. . . . . . . . . . 11
| |
| 7 | 6 | 3ad2ant1 1021 |
. . . . . . . . . 10
|
| 8 | 7 | adantr 276 |
. . . . . . . . 9
|
| 9 | 8 | ad3antrrr 492 |
. . . . . . . 8
|
| 10 | simpr 110 |
. . . . . . . . . . . . . . 15
| |
| 11 | simpllr 534 |
. . . . . . . . . . . . . . 15
| |
| 12 | 10, 11 | breq12d 4064 |
. . . . . . . . . . . . . 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 8696 |
. . . . . . . . . . . . . . 15
#
#
# | |
| 20 | 13, 14, 16, 18, 19 | syl22anc 1251 |
. . . . . . . . . . . . . 14
#
#
# |
| 21 | 12, 20 | bitrd 188 |
. . . . . . . . . . . . 13
#
# # |
| 22 | simprl 529 |
. . . . . . . . . . . . . . . . 17
| |
| 23 | 22 | ad2antrr 488 |
. . . . . . . . . . . . . . . 16
|
| 24 | 23 | ad3antrrr 492 |
. . . . . . . . . . . . . . 15
|
| 25 | reapcotr 8691 |
. . . . . . . . . . . . . . 15
# # # | |
| 26 | 13, 16, 24, 25 | syl3anc 1250 |
. . . . . . . . . . . . . 14
# # # |
| 27 | simprr 531 |
. . . . . . . . . . . . . . . . 17
| |
| 28 | 27 | ad2antrr 488 |
. . . . . . . . . . . . . . . 16
|
| 29 | 28 | ad3antrrr 492 |
. . . . . . . . . . . . . . 15
|
| 30 | reapcotr 8691 |
. . . . . . . . . . . . . . 15
# # # | |
| 31 | 14, 18, 29, 30 | syl3anc 1250 |
. . . . . . . . . . . . . 14
# # # |
| 32 | 26, 31 | orim12d 788 |
. . . . . . . . . . . . 13
# # # #
# # |
| 33 | 21, 32 | sylbid 150 |
. . . . . . . . . . . 12
#
# # # # |
| 34 | or4 773 |
. . . . . . . . . . . 12
# # # # # #
# # | |
| 35 | 33, 34 | imbitrdi 161 |
. . . . . . . . . . 11
#
# # # # |
| 36 | simplr 528 |
. . . . . . . . . . . . . . 15
| |
| 37 | 36 | ad3antrrr 492 |
. . . . . . . . . . . . . 14
|
| 38 | 10, 37 | breq12d 4064 |
. . . . . . . . . . . . 13
#
# |
| 39 | apreim 8696 |
. . . . . . . . . . . . . 14
#
#
# | |
| 40 | 13, 14, 24, 29, 39 | syl22anc 1251 |
. . . . . . . . . . . . 13
#
#
# |
| 41 | 38, 40 | bitrd 188 |
. . . . . . . . . . . 12
#
# #
|
| 42 | 11, 37 | breq12d 4064 |
. . . . . . . . . . . . 13
#
# |
| 43 | apreim 8696 |
. . . . . . . . . . . . . 14
#
#
# | |
| 44 | 16, 18, 24, 29, 43 | syl22anc 1251 |
. . . . . . . . . . . . 13
#
#
# |
| 45 | 42, 44 | bitrd 188 |
. . . . . . . . . . . 12
#
# # |
| 46 | 41, 45 | orbi12d 795 |
. . . . . . . . . . 11
# #
# # # # |
| 47 | 35, 46 | sylibrd 169 |
. . . . . . . . . 10
#
# # |
| 48 | 47 | ex 115 |
. . . . . . . . 9
#
# # |
| 49 | 48 | rexlimdvva 2632 |
. . . . . . . 8
#
# # |
| 50 | 9, 49 | mpd 13 |
. . . . . . 7
#
# # |
| 51 | 50 | ex 115 |
. . . . . 6
#
# # |
| 52 | 51 | rexlimdvva 2632 |
. . . . 5
#
# # |
| 53 | 5, 52 | mpd 13 |
. . . 4
#
# # |
| 54 | 53 | ex 115 |
. . 3
#
# # |
| 55 | 54 | rexlimdvva 2632 |
. 2
#
# # |
| 56 | 2, 55 | mpd 13 |
1
#
# # |
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 104
wb 105 wo 710 w3a 981 wceq 1373 wcel 2177 wrex 2486 class class class wbr 4051
(class class class)co 5957 cc 7943 cr 7944 ci 7947 caddc 7948 cmul 7950 # cap 8674 |
| 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 615 ax-in2 616 ax-io 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2179 ax-14 2180 ax-ext 2188 ax-sep 4170 ax-pow 4226 ax-pr 4261 ax-un 4488 ax-setind 4593 ax-cnex 8036 ax-resscn 8037 ax-1cn 8038 ax-1re 8039 ax-icn 8040 ax-addcl 8041 ax-addrcl 8042 ax-mulcl 8043 ax-mulrcl 8044 ax-addcom 8045 ax-mulcom 8046 ax-addass 8047 ax-mulass 8048 ax-distr 8049 ax-i2m1 8050 ax-0lt1 8051 ax-1rid 8052 ax-0id 8053 ax-rnegex 8054 ax-precex 8055 ax-cnre 8056 ax-pre-ltirr 8057 ax-pre-ltwlin 8058 ax-pre-lttrn 8059 ax-pre-apti 8060 ax-pre-ltadd 8061 ax-pre-mulgt0 8062 |
| This theorem depends on definitions: df-bi 117 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2193 df-cleq 2199 df-clel 2202 df-nfc 2338 df-ne 2378 df-nel 2473 df-ral 2490 df-rex 2491 df-reu 2492 df-rab 2494 df-v 2775 df-sbc 3003 df-dif 3172 df-un 3174 df-in 3176 df-ss 3183 df-pw 3623 df-sn 3644 df-pr 3645 df-op 3647 df-uni 3857 df-br 4052 df-opab 4114 df-id 4348 df-xp 4689 df-rel 4690 df-cnv 4691 df-co 4692 df-dm 4693 df-iota 5241 df-fun 5282 df-fv 5288 df-riota 5912 df-ov 5960 df-oprab 5961 df-mpo 5962 df-pnf 8129 df-mnf 8130 df-ltxr 8132 df-sub 8265 df-neg 8266 df-reap 8668 df-ap 8675 |
| This theorem is referenced by: addext 8703 mulext 8707 aptap 8743 mul0eqap 8763 |
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