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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 2046 |
. . . . . 6
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2 | 1 | rexbidv 2327 |
. . . . 5
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3 | 2 | cbvabv 2161 |
. . . 4
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4 | 3 | sseq1i 2969 |
. . 3
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5 | r19.23v 2425 |
. . . . 5
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6 | 5 | albii 1359 |
. . . 4
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7 | ralcom4 2576 |
. . . 4
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8 | abss 3009 |
. . . 4
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9 | 6, 7, 8 | 3bitr4i 201 |
. . 3
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10 | 4, 9 | bitr4i 176 |
. 2
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11 | nfv 1421 |
. . . . 5
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12 | eleq1 2100 |
. . . . 5
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13 | 11, 12 | ceqsalg 2582 |
. . . 4
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14 | 13 | ralimi 2384 |
. . 3
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15 | ralbi 2445 |
. . 3
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16 | 14, 15 | syl 14 |
. 2
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17 | 10, 16 | syl5rbb 182 |
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 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-in 2924 df-ss 2931 |
This theorem is referenced by: (None) |
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