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Mirrors > Home > ILE Home > Th. List > xpiindim | Unicode version |
Description: Distributive law for cross product over indexed intersection. (Contributed by Jim Kingdon, 7-Dec-2018.) |
Ref | Expression |
---|---|
xpiindim |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relxp 4390 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
2 | 1 | rgenw 2370 |
. . . . 5
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3 | eleq1 2097 |
. . . . . . 7
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4 | 3 | cbvexv 1792 |
. . . . . 6
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5 | r19.2m 3303 |
. . . . . 6
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6 | 4, 5 | sylanbr 269 |
. . . . 5
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7 | 2, 6 | mpan2 401 |
. . . 4
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8 | reliin 4402 |
. . . 4
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9 | 7, 8 | syl 14 |
. . 3
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10 | relxp 4390 |
. . 3
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11 | 9, 10 | jctil 295 |
. 2
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12 | r19.28mv 3308 |
. . . . . . 7
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13 | 4, 12 | sylbir 125 |
. . . . . 6
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14 | 13 | bicomd 129 |
. . . . 5
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15 | vex 2554 |
. . . . . . 7
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16 | eliin 3653 |
. . . . . . 7
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17 | 15, 16 | ax-mp 7 |
. . . . . 6
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18 | 17 | anbi2i 430 |
. . . . 5
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19 | opelxp 4317 |
. . . . . 6
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20 | 19 | ralbii 2324 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
21 | 14, 18, 20 | 3bitr4g 212 |
. . . 4
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22 | opelxp 4317 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
23 | vex 2554 |
. . . . . 6
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24 | 23, 15 | opex 3957 |
. . . . 5
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25 | eliin 3653 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
26 | 24, 25 | ax-mp 7 |
. . . 4
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27 | 21, 22, 26 | 3bitr4g 212 |
. . 3
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28 | 27 | eqrelrdv2 4382 |
. 2
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29 | 11, 28 | mpancom 399 |
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 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-iin 3651 df-opab 3810 df-xp 4294 df-rel 4295 |
This theorem is referenced by: xpriindim 4417 |
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