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Mirrors > Home > ILE Home > Th. List > xpiundi | Unicode version |
Description: Distributive law for cross product over indexed union. (Contributed by Mario Carneiro, 27-Apr-2014.) |
Ref | Expression |
---|---|
xpiundi |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexcom 2598 |
. . . 4
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2 | eliun 3825 |
. . . . . . . 8
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3 | 2 | anbi1i 454 |
. . . . . . 7
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4 | 3 | exbii 1585 |
. . . . . 6
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5 | df-rex 2423 |
. . . . . 6
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6 | df-rex 2423 |
. . . . . . . 8
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7 | 6 | rexbii 2445 |
. . . . . . 7
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8 | rexcom4 2712 |
. . . . . . 7
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9 | r19.41v 2590 |
. . . . . . . 8
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10 | 9 | exbii 1585 |
. . . . . . 7
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11 | 7, 8, 10 | 3bitri 205 |
. . . . . 6
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12 | 4, 5, 11 | 3bitr4i 211 |
. . . . 5
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13 | 12 | rexbii 2445 |
. . . 4
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14 | elxp2 4565 |
. . . . 5
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15 | 14 | rexbii 2445 |
. . . 4
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16 | 1, 13, 15 | 3bitr4i 211 |
. . 3
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17 | elxp2 4565 |
. . 3
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18 | eliun 3825 |
. . 3
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19 | 16, 17, 18 | 3bitr4i 211 |
. 2
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20 | 19 | eqriv 2137 |
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 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-iun 3823 df-opab 3998 df-xp 4553 |
This theorem is referenced by: xpexgALT 6039 txbasval 12475 |
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