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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 2478 |
. . . 4
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2 | vex 2622 |
. . . . . 6
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3 | eliin 3735 |
. . . . . 6
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4 | 2, 3 | ax-mp 7 |
. . . . 5
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5 | 4 | rexbii 2385 |
. . . 4
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6 | eliun 3734 |
. . . . 5
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7 | 6 | ralbii 2384 |
. . . 4
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8 | 1, 5, 7 | 3imtr4i 199 |
. . 3
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9 | eliun 3734 |
. . 3
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10 | eliin 3735 |
. . . 4
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11 | 2, 10 | ax-mp 7 |
. . 3
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12 | 8, 9, 11 | 3imtr4i 199 |
. 2
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13 | 12 | ssriv 3029 |
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-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 |
This theorem depends on definitions: df-bi 115 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 df-rex 2365 df-v 2621 df-in 3005 df-ss 3012 df-iun 3732 df-iin 3733 |
This theorem is referenced by: (None) |
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