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Mirrors > Home > ILE Home > Th. List > djuss | Unicode version |
Description: A disjoint union is a subset of a Cartesian product. (Contributed by AV, 25-Jun-2022.) |
Ref | Expression |
---|---|
djuss |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | djur 6903 |
. . 3
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2 | simpr 109 |
. . . . . . 7
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3 | df-inl 6881 |
. . . . . . . . 9
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4 | opeq2 3670 |
. . . . . . . . 9
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5 | elex 2666 |
. . . . . . . . 9
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6 | 0ex 4013 |
. . . . . . . . . . 11
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7 | vex 2658 |
. . . . . . . . . . 11
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8 | 6, 7 | opex 4109 |
. . . . . . . . . 10
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9 | 8 | a1i 9 |
. . . . . . . . 9
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10 | 3, 4, 5, 9 | fvmptd3 5466 |
. . . . . . . 8
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11 | 10 | adantr 272 |
. . . . . . 7
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12 | 2, 11 | eqtrd 2145 |
. . . . . 6
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13 | elun1 3207 |
. . . . . . . . 9
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14 | 6 | prid1 3593 |
. . . . . . . . 9
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15 | 13, 14 | jctil 308 |
. . . . . . . 8
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16 | 15 | adantr 272 |
. . . . . . 7
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17 | opelxp 4527 |
. . . . . . 7
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18 | 16, 17 | sylibr 133 |
. . . . . 6
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19 | 12, 18 | eqeltrd 2189 |
. . . . 5
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20 | 19 | rexlimiva 2516 |
. . . 4
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21 | simpr 109 |
. . . . . . 7
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22 | df-inr 6882 |
. . . . . . . . 9
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23 | opeq2 3670 |
. . . . . . . . 9
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24 | elex 2666 |
. . . . . . . . 9
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25 | 1oex 6272 |
. . . . . . . . . . 11
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26 | 25, 7 | opex 4109 |
. . . . . . . . . 10
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27 | 26 | a1i 9 |
. . . . . . . . 9
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28 | 22, 23, 24, 27 | fvmptd3 5466 |
. . . . . . . 8
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29 | 28 | adantr 272 |
. . . . . . 7
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30 | 21, 29 | eqtrd 2145 |
. . . . . 6
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31 | elun2 3208 |
. . . . . . . . 9
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32 | 31 | adantr 272 |
. . . . . . . 8
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33 | 25 | prid2 3594 |
. . . . . . . 8
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34 | 32, 33 | jctil 308 |
. . . . . . 7
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35 | opelxp 4527 |
. . . . . . 7
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36 | 34, 35 | sylibr 133 |
. . . . . 6
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37 | 30, 36 | eqeltrd 2189 |
. . . . 5
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38 | 37 | rexlimiva 2516 |
. . . 4
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39 | 20, 38 | jaoi 688 |
. . 3
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40 | 1, 39 | sylbi 120 |
. 2
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41 | 40 | ssriv 3065 |
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 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-13 1472 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-sep 4004 ax-nul 4012 ax-pow 4056 ax-pr 4089 ax-un 4313 |
This theorem depends on definitions: df-bi 116 df-3an 945 df-tru 1315 df-nf 1418 df-sb 1717 df-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ral 2393 df-rex 2394 df-v 2657 df-sbc 2877 df-csb 2970 df-dif 3037 df-un 3039 df-in 3041 df-ss 3048 df-nul 3328 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-br 3894 df-opab 3948 df-mpt 3949 df-tr 3985 df-id 4173 df-iord 4246 df-on 4248 df-suc 4251 df-xp 4503 df-rel 4504 df-cnv 4505 df-co 4506 df-dm 4507 df-rn 4508 df-res 4509 df-iota 5044 df-fun 5081 df-fn 5082 df-f 5083 df-f1 5084 df-fo 5085 df-f1o 5086 df-fv 5087 df-1st 5989 df-2nd 5990 df-1o 6264 df-dju 6872 df-inl 6881 df-inr 6882 |
This theorem is referenced by: eldju1st 6905 |
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