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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 6664 |
. . 3
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2 | simpr 108 |
. . . . . . 7
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3 | df-inl 6646 |
. . . . . . . . . 10
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4 | 3 | a1i 9 |
. . . . . . . . 9
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5 | opeq2 3597 |
. . . . . . . . . 10
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6 | 5 | adantl 271 |
. . . . . . . . 9
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7 | elex 2621 |
. . . . . . . . 9
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8 | 0ex 3931 |
. . . . . . . . . . 11
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9 | vex 2615 |
. . . . . . . . . . 11
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10 | 8, 9 | opex 4020 |
. . . . . . . . . 10
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11 | 10 | a1i 9 |
. . . . . . . . 9
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12 | 4, 6, 7, 11 | fvmptd 5330 |
. . . . . . . 8
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13 | 12 | adantr 270 |
. . . . . . 7
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14 | 2, 13 | eqtrd 2115 |
. . . . . 6
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15 | elun1 3151 |
. . . . . . . . 9
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16 | 8 | prid1 3522 |
. . . . . . . . 9
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17 | 15, 16 | jctil 305 |
. . . . . . . 8
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18 | 17 | adantr 270 |
. . . . . . 7
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19 | opelxp 4430 |
. . . . . . 7
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20 | 18, 19 | sylibr 132 |
. . . . . 6
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21 | 14, 20 | eqeltrd 2159 |
. . . . 5
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22 | 21 | rexlimiva 2478 |
. . . 4
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23 | simpr 108 |
. . . . . . 7
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24 | df-inr 6647 |
. . . . . . . . . 10
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25 | 24 | a1i 9 |
. . . . . . . . 9
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26 | opeq2 3597 |
. . . . . . . . . 10
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27 | 26 | adantl 271 |
. . . . . . . . 9
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28 | elex 2621 |
. . . . . . . . 9
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29 | 1oex 6121 |
. . . . . . . . . . 11
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30 | 29, 9 | opex 4020 |
. . . . . . . . . 10
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31 | 30 | a1i 9 |
. . . . . . . . 9
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32 | 25, 27, 28, 31 | fvmptd 5330 |
. . . . . . . 8
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33 | 32 | adantr 270 |
. . . . . . 7
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34 | 23, 33 | eqtrd 2115 |
. . . . . 6
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35 | elun2 3152 |
. . . . . . . . 9
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36 | 35 | adantr 270 |
. . . . . . . 8
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37 | 29 | prid2 3523 |
. . . . . . . 8
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38 | 36, 37 | jctil 305 |
. . . . . . 7
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39 | opelxp 4430 |
. . . . . . 7
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40 | 38, 39 | sylibr 132 |
. . . . . 6
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41 | 34, 40 | eqeltrd 2159 |
. . . . 5
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42 | 41 | rexlimiva 2478 |
. . . 4
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43 | 22, 42 | jaoi 669 |
. . 3
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44 | 1, 43 | syl 14 |
. 2
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45 | 44 | ssriv 3014 |
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 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3922 ax-nul 3930 ax-pow 3974 ax-pr 4000 ax-un 4224 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ral 2358 df-rex 2359 df-v 2614 df-sbc 2827 df-csb 2920 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-nul 3270 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-int 3663 df-br 3812 df-opab 3866 df-mpt 3867 df-tr 3902 df-id 4084 df-iord 4157 df-on 4159 df-suc 4162 df-iom 4369 df-xp 4407 df-rel 4408 df-cnv 4409 df-co 4410 df-dm 4411 df-rn 4412 df-iota 4934 df-fun 4971 df-fn 4972 df-f 4973 df-fo 4975 df-fv 4977 df-1st 5846 df-2nd 5847 df-1o 6113 df-dju 6638 df-inl 6646 df-inr 6647 |
This theorem is referenced by: eldju1st 6669 |
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