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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 2196 |
. . . . . 6
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2 | 1 | rexbidv 2491 |
. . . . 5
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3 | 2 | cbvabv 2314 |
. . . 4
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4 | 3 | sseq1i 3196 |
. . 3
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5 | r19.23v 2599 |
. . . . 5
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6 | 5 | albii 1481 |
. . . 4
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7 | ralcom4 2774 |
. . . 4
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8 | abss 3239 |
. . . 4
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9 | 6, 7, 8 | 3bitr4i 212 |
. . 3
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10 | 4, 9 | bitr4i 187 |
. 2
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11 | nfv 1539 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() | |
12 | eleq1 2252 |
. . . . 5
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13 | 11, 12 | ceqsalg 2780 |
. . . 4
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14 | 13 | ralimi 2553 |
. . 3
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15 | ralbi 2622 |
. . 3
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16 | 14, 15 | syl 14 |
. 2
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17 | 10, 16 | bitr2id 193 |
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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-in 3150 df-ss 3157 |
This theorem is referenced by: iunopn 13905 |
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