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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 7062 |
. . 3
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2 | simpr 110 |
. . . . . . 7
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3 | df-inl 7040 |
. . . . . . . . 9
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4 | opeq2 3777 |
. . . . . . . . 9
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5 | elex 2748 |
. . . . . . . . 9
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6 | 0ex 4127 |
. . . . . . . . . . 11
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7 | vex 2740 |
. . . . . . . . . . 11
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8 | 6, 7 | opex 4226 |
. . . . . . . . . 10
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9 | 8 | a1i 9 |
. . . . . . . . 9
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10 | 3, 4, 5, 9 | fvmptd3 5605 |
. . . . . . . 8
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11 | 10 | adantr 276 |
. . . . . . 7
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12 | 2, 11 | eqtrd 2210 |
. . . . . 6
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13 | elun1 3302 |
. . . . . . . . 9
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14 | 6 | prid1 3697 |
. . . . . . . . 9
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15 | 13, 14 | jctil 312 |
. . . . . . . 8
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16 | 15 | adantr 276 |
. . . . . . 7
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17 | opelxp 4653 |
. . . . . . 7
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18 | 16, 17 | sylibr 134 |
. . . . . 6
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19 | 12, 18 | eqeltrd 2254 |
. . . . 5
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20 | 19 | rexlimiva 2589 |
. . . 4
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21 | simpr 110 |
. . . . . . 7
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22 | df-inr 7041 |
. . . . . . . . 9
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23 | opeq2 3777 |
. . . . . . . . 9
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24 | elex 2748 |
. . . . . . . . 9
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25 | 1oex 6419 |
. . . . . . . . . . 11
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26 | 25, 7 | opex 4226 |
. . . . . . . . . 10
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27 | 26 | a1i 9 |
. . . . . . . . 9
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28 | 22, 23, 24, 27 | fvmptd3 5605 |
. . . . . . . 8
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29 | 28 | adantr 276 |
. . . . . . 7
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30 | 21, 29 | eqtrd 2210 |
. . . . . 6
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31 | elun2 3303 |
. . . . . . . . 9
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32 | 31 | adantr 276 |
. . . . . . . 8
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33 | 25 | prid2 3698 |
. . . . . . . 8
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34 | 32, 33 | jctil 312 |
. . . . . . 7
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35 | opelxp 4653 |
. . . . . . 7
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36 | 34, 35 | sylibr 134 |
. . . . . 6
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37 | 30, 36 | eqeltrd 2254 |
. . . . 5
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38 | 37 | rexlimiva 2589 |
. . . 4
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39 | 20, 38 | jaoi 716 |
. . 3
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40 | 1, 39 | sylbi 121 |
. 2
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41 | 40 | ssriv 3159 |
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-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4206 ax-un 4430 |
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-ral 2460 df-rex 2461 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-id 4290 df-iord 4363 df-on 4365 df-suc 4368 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-fv 5220 df-1st 6135 df-2nd 6136 df-1o 6411 df-dju 7031 df-inl 7040 df-inr 7041 |
This theorem is referenced by: eldju1st 7064 |
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