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Mirrors > Home > ILE Home > Th. List > djulclb | Unicode version |
Description: Left biconditional closure of disjoint union. (Contributed by Jim Kingdon, 2-Jul-2022.) |
Ref | Expression |
---|---|
djulclb |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | djulcl 7049 |
. 2
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2 | 1n0 6432 |
. . . . . . . . . 10
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3 | 2 | necomi 2432 |
. . . . . . . . 9
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4 | 0ex 4130 |
. . . . . . . . . 10
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5 | 4 | elsn 3608 |
. . . . . . . . 9
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6 | 3, 5 | nemtbir 2436 |
. . . . . . . 8
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7 | 6 | intnanr 930 |
. . . . . . 7
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8 | opelxp 4656 |
. . . . . . 7
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9 | 7, 8 | mtbir 671 |
. . . . . 6
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10 | elex 2748 |
. . . . . . . . . . . 12
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11 | opexg 4228 |
. . . . . . . . . . . . 13
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12 | 4, 11 | mpan 424 |
. . . . . . . . . . . 12
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13 | opeq2 3779 |
. . . . . . . . . . . . 13
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14 | df-inl 7045 |
. . . . . . . . . . . . 13
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15 | 13, 14 | fvmptg 5592 |
. . . . . . . . . . . 12
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16 | 10, 12, 15 | syl2anc 411 |
. . . . . . . . . . 11
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17 | 16 | adantr 276 |
. . . . . . . . . 10
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18 | df-dju 7036 |
. . . . . . . . . . . . 13
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19 | 18 | eleq2i 2244 |
. . . . . . . . . . . 12
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20 | 19 | biimpi 120 |
. . . . . . . . . . 11
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21 | 20 | adantl 277 |
. . . . . . . . . 10
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22 | 17, 21 | eqeltrrd 2255 |
. . . . . . . . 9
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23 | elun 3276 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
24 | 22, 23 | sylib 122 |
. . . . . . . 8
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25 | 24 | orcomd 729 |
. . . . . . 7
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26 | 25 | ord 724 |
. . . . . 6
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27 | 9, 26 | mpi 15 |
. . . . 5
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28 | opelxp 4656 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
29 | 27, 28 | sylib 122 |
. . . 4
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30 | 29 | simprd 114 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
31 | 30 | ex 115 |
. 2
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32 | 1, 31 | impbid2 143 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-suc 4371 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-iota 5178 df-fun 5218 df-fv 5224 df-1o 6416 df-dju 7036 df-inl 7045 |
This theorem is referenced by: exmidfodomrlemr 7200 |
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