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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 8312 |
. . 3
      | |
| 2 | 1 | 3ad2ant3 1051 |
. 2
|
| 3 | cnre 8312 |
. . . . . . 7
| |
| 4 | 3 | 3ad2ant2 1050 |
. . . . . 6
|
| 5 | 4 | ad2antrr 492 |
. . . . 5
|
| 6 | cnre 8312 |
. . . . . . . . . . 11
| |
| 7 | 6 | 3ad2ant1 1049 |
. . . . . . . . . 10
|
| 8 | 7 | adantr 276 |
. . . . . . . . 9
|
| 9 | 8 | ad3antrrr 496 |
. . . . . . . 8
|
| 10 | simpr 110 |
. . . . . . . . . . . . . . 15
| |
| 11 | simpllr 540 |
. . . . . . . . . . . . . . 15
| |
| 12 | 10, 11 | breq12d 4138 |
. . . . . . . . . . . . . 14
#
# |
| 13 | simplrl 541 |
. . . . . . . . . . . . . . 15
| |
| 14 | simplrr 542 |
. . . . . . . . . . . . . . 15
| |
| 15 | simprl 535 |
. . . . . . . . . . . . . . . 16
| |
| 16 | 15 | ad3antrrr 496 |
. . . . . . . . . . . . . . 15
|
| 17 | simprr 537 |
. . . . . . . . . . . . . . . 16
| |
| 18 | 17 | ad3antrrr 496 |
. . . . . . . . . . . . . . 15
|
| 19 | apreim 8921 |
. . . . . . . . . . . . . . 15
#
#
# | |
| 20 | 13, 14, 16, 18, 19 | syl22anc 1279 |
. . . . . . . . . . . . . 14
#
#
# |
| 21 | 12, 20 | bitrd 188 |
. . . . . . . . . . . . 13
#
# # |
| 22 | simprl 535 |
. . . . . . . . . . . . . . . . 17
| |
| 23 | 22 | ad2antrr 492 |
. . . . . . . . . . . . . . . 16
|
| 24 | 23 | ad3antrrr 496 |
. . . . . . . . . . . . . . 15
|
| 25 | reapcotr 8916 |
. . . . . . . . . . . . . . 15
# # # | |
| 26 | 13, 16, 24, 25 | syl3anc 1278 |
. . . . . . . . . . . . . 14
# # # |
| 27 | simprr 537 |
. . . . . . . . . . . . . . . . 17
| |
| 28 | 27 | ad2antrr 492 |
. . . . . . . . . . . . . . . 16
|
| 29 | 28 | ad3antrrr 496 |
. . . . . . . . . . . . . . 15
|
| 30 | reapcotr 8916 |
. . . . . . . . . . . . . . 15
# # # | |
| 31 | 14, 18, 29, 30 | syl3anc 1278 |
. . . . . . . . . . . . . 14
# # # |
| 32 | 26, 31 | orim12d 798 |
. . . . . . . . . . . . 13
# # # #
# # |
| 33 | 21, 32 | sylbid 150 |
. . . . . . . . . . . 12
#
# # # # |
| 34 | or4 783 |
. . . . . . . . . . . 12
# # # # # #
# # | |
| 35 | 33, 34 | imbitrdi 161 |
. . . . . . . . . . 11
#
# # # # |
| 36 | simplr 533 |
. . . . . . . . . . . . . . 15
| |
| 37 | 36 | ad3antrrr 496 |
. . . . . . . . . . . . . 14
|
| 38 | 10, 37 | breq12d 4138 |
. . . . . . . . . . . . 13
#
# |
| 39 | apreim 8921 |
. . . . . . . . . . . . . 14
#
#
# | |
| 40 | 13, 14, 24, 29, 39 | syl22anc 1279 |
. . . . . . . . . . . . 13
#
#
# |
| 41 | 38, 40 | bitrd 188 |
. . . . . . . . . . . 12
#
# #
|
| 42 | 11, 37 | breq12d 4138 |
. . . . . . . . . . . . 13
#
# |
| 43 | apreim 8921 |
. . . . . . . . . . . . . 14
#
#
# | |
| 44 | 16, 18, 24, 29, 43 | syl22anc 1279 |
. . . . . . . . . . . . 13
#
#
# |
| 45 | 42, 44 | bitrd 188 |
. . . . . . . . . . . 12
#
# # |
| 46 | 41, 45 | orbi12d 805 |
. . . . . . . . . . 11
# #
# # # # |
| 47 | 35, 46 | sylibrd 169 |
. . . . . . . . . 10
#
# # |
| 48 | 47 | ex 115 |
. . . . . . . . 9
#
# # |
| 49 | 48 | rexlimdvva 2676 |
. . . . . . . 8
#
# # |
| 50 | 9, 49 | mpd 13 |
. . . . . . 7
#
# # |
| 51 | 50 | ex 115 |
. . . . . 6
#
# # |
| 52 | 51 | rexlimdvva 2676 |
. . . . 5
#
# # |
| 53 | 5, 52 | mpd 13 |
. . . 4
#
# # |
| 54 | 53 | ex 115 |
. . 3
#
# # |
| 55 | 54 | rexlimdvva 2676 |
. 2
#
# # |
| 56 | 2, 55 | mpd 13 |
1
#
# # |
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 104
wb 105 wo 720 w3a 1009 wceq 1402 wcel 2209 wrex 2529 class class class wbr 4125
(class class class)co 6075 cc 8167 cr 8168 ci 8171 caddc 8172 cmul 8174 # cap 8899 |
| 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 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4244 ax-pow 4306 ax-pr 4341 ax-un 4573 ax-setind 4679 ax-cnex 8260 ax-resscn 8261 ax-1cn 8262 ax-1re 8263 ax-icn 8264 ax-addcl 8265 ax-addrcl 8266 ax-mulcl 8267 ax-mulrcl 8268 ax-addcom 8269 ax-mulcom 8270 ax-addass 8271 ax-mulass 8272 ax-distr 8273 ax-i2m1 8274 ax-0lt1 8275 ax-1rid 8276 ax-0id 8277 ax-rnegex 8278 ax-precex 8279 ax-cnre 8280 ax-pre-ltirr 8281 ax-pre-ltwlin 8282 ax-pre-lttrn 8283 ax-pre-apti 8284 ax-pre-ltadd 8285 ax-pre-mulgt0 8286 |
| This theorem depends on definitions: df-bi 117 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rab 2537 df-v 2823 df-sbc 3052 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-pw 3687 df-sn 3711 df-pr 3712 df-op 3714 df-uni 3931 df-br 4126 df-opab 4188 df-id 4433 df-xp 4775 df-rel 4776 df-cnv 4777 df-co 4778 df-dm 4779 df-iota 5332 df-fun 5374 df-fv 5380 df-riota 6028 df-ov 6078 df-oprab 6079 df-mpo 6080 df-pnf 8352 df-mnf 8353 df-ltxr 8355 df-sub 8489 df-neg 8490 df-reap 8893 df-ap 8900 |
| This theorem is referenced by: addext 8928 mulext 8932 aptap 8968 mul0eqap 8990 |
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