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Mirrors > Home > ILE Home > Th. List > iuniin | Unicode version |
Description: Law combining indexed union with indexed intersection. Eq. 14 in [KuratowskiMostowski] p. 109. This theorem also appears as the last example at http://en.wikipedia.org/wiki/Union%5F%28set%5Ftheory%29. (Contributed by NM, 17-Aug-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
Ref | Expression |
---|---|
iuniin |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.12 2583 |
. . . 4
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2 | vex 2740 |
. . . . . 6
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3 | eliin 3891 |
. . . . . 6
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4 | 2, 3 | ax-mp 5 |
. . . . 5
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5 | 4 | rexbii 2484 |
. . . 4
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6 | eliun 3890 |
. . . . 5
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7 | 6 | ralbii 2483 |
. . . 4
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8 | 1, 5, 7 | 3imtr4i 201 |
. . 3
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9 | eliun 3890 |
. . 3
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10 | eliin 3891 |
. . . 4
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11 | 2, 10 | ax-mp 5 |
. . 3
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12 | 8, 9, 11 | 3imtr4i 201 |
. 2
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13 | 12 | 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-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-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-in 3135 df-ss 3142 df-iun 3888 df-iin 3889 |
This theorem is referenced by: (None) |
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