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Mirrors > Home > ILE Home > Th. List > diftpsn3 | Unicode version |
Description: Removal of a singleton from an unordered triple. (Contributed by Alexander van der Vekens, 5-Oct-2017.) |
Ref | Expression |
---|---|
diftpsn3 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-tp 3383 |
. . . 4
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2 | 1 | a1i 9 |
. . 3
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3 | 2 | difeq1d 3061 |
. 2
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4 | difundir 3190 |
. . 3
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5 | 4 | a1i 9 |
. 2
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6 | df-pr 3382 |
. . . . . . . . 9
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7 | 6 | a1i 9 |
. . . . . . . 8
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8 | 7 | ineq1d 3137 |
. . . . . . 7
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9 | incom 3129 |
. . . . . . . . 9
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10 | indi 3184 |
. . . . . . . . 9
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11 | 9, 10 | eqtri 2060 |
. . . . . . . 8
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12 | 11 | a1i 9 |
. . . . . . 7
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13 | necom 2289 |
. . . . . . . . . . 11
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14 | disjsn2 3433 |
. . . . . . . . . . 11
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15 | 13, 14 | sylbi 114 |
. . . . . . . . . 10
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16 | 15 | adantr 261 |
. . . . . . . . 9
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17 | necom 2289 |
. . . . . . . . . . 11
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18 | disjsn2 3433 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
19 | 17, 18 | sylbi 114 |
. . . . . . . . . 10
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20 | 19 | adantl 262 |
. . . . . . . . 9
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21 | 16, 20 | uneq12d 3098 |
. . . . . . . 8
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22 | unidm 3086 |
. . . . . . . 8
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23 | 21, 22 | syl6eq 2088 |
. . . . . . 7
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24 | 8, 12, 23 | 3eqtrd 2076 |
. . . . . 6
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25 | disj3 3272 |
. . . . . 6
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26 | 24, 25 | sylib 127 |
. . . . 5
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27 | 26 | eqcomd 2045 |
. . . 4
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28 | difid 3292 |
. . . . 5
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29 | 28 | a1i 9 |
. . . 4
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30 | 27, 29 | uneq12d 3098 |
. . 3
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31 | un0 3251 |
. . 3
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32 | 30, 31 | syl6eq 2088 |
. 2
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33 | 3, 5, 32 | 3eqtrd 2076 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rab 2315 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-sn 3381 df-pr 3382 df-tp 3383 |
This theorem is referenced by: (None) |
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