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Mirrors > Home > ILE Home > Th. List > uniiunlem | Unicode version |
Description: A subset relationship useful for converting union to indexed union using dfiun2 or dfiun2g and intersection to indexed intersection using dfiin2 . (Contributed by NM, 5-Oct-2006.) (Proof shortened by Mario Carneiro, 26-Sep-2015.) |
Ref | Expression |
---|---|
uniiunlem |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq1 2147 |
. . . . . 6
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2 | 1 | rexbidv 2439 |
. . . . 5
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3 | 2 | cbvabv 2265 |
. . . 4
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4 | 3 | sseq1i 3128 |
. . 3
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5 | r19.23v 2544 |
. . . . 5
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6 | 5 | albii 1447 |
. . . 4
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7 | ralcom4 2711 |
. . . 4
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8 | abss 3171 |
. . . 4
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9 | 6, 7, 8 | 3bitr4i 211 |
. . 3
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10 | 4, 9 | bitr4i 186 |
. 2
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11 | nfv 1509 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() | |
12 | eleq1 2203 |
. . . . 5
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13 | 11, 12 | ceqsalg 2717 |
. . . 4
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14 | 13 | ralimi 2498 |
. . 3
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15 | ralbi 2567 |
. . 3
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16 | 14, 15 | syl 14 |
. 2
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17 | 10, 16 | syl5rbb 192 |
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-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 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-in 3082 df-ss 3089 |
This theorem is referenced by: iunopn 12208 |
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