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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 2194 |
. . . . . 6
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2 | 1 | rexbidv 2488 |
. . . . 5
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3 | 2 | cbvabv 2312 |
. . . 4
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4 | 3 | sseq1i 3193 |
. . 3
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5 | r19.23v 2596 |
. . . . 5
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6 | 5 | albii 1480 |
. . . 4
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7 | ralcom4 2771 |
. . . 4
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8 | abss 3236 |
. . . 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 1538 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() | |
12 | eleq1 2250 |
. . . . 5
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13 | 11, 12 | ceqsalg 2777 |
. . . 4
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14 | 13 | ralimi 2550 |
. . 3
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15 | ralbi 2619 |
. . 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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2169 |
This theorem depends on definitions: df-bi 117 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-v 2751 df-in 3147 df-ss 3154 |
This theorem is referenced by: iunopn 13742 |
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