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Mirrors > Home > ILE Home > Th. List > unissb | Unicode version |
Description: Relationship involving membership, subset, and union. Exercise 5 of [Enderton] p. 26 and its converse. (Contributed by NM, 20-Sep-2003.) |
Ref | Expression |
---|---|
unissb |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluni 3827 |
. . . . . 6
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2 | 1 | imbi1i 238 |
. . . . 5
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3 | 19.23v 1894 |
. . . . 5
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4 | 2, 3 | bitr4i 187 |
. . . 4
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5 | 4 | albii 1481 |
. . 3
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6 | alcom 1489 |
. . . 4
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7 | 19.21v 1884 |
. . . . . 6
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8 | impexp 263 |
. . . . . . . 8
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9 | bi2.04 248 |
. . . . . . . 8
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10 | 8, 9 | bitri 184 |
. . . . . . 7
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11 | 10 | albii 1481 |
. . . . . 6
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12 | dfss2 3159 |
. . . . . . 7
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13 | 12 | imbi2i 226 |
. . . . . 6
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14 | 7, 11, 13 | 3bitr4i 212 |
. . . . 5
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15 | 14 | albii 1481 |
. . . 4
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16 | 6, 15 | bitri 184 |
. . 3
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17 | 5, 16 | bitri 184 |
. 2
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18 | dfss2 3159 |
. 2
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19 | df-ral 2473 |
. 2
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20 | 17, 18, 19 | 3bitr4i 212 |
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-v 2754 df-in 3150 df-ss 3157 df-uni 3825 |
This theorem is referenced by: uniss2 3855 ssunieq 3857 sspwuni 3986 pwssb 3987 bm2.5ii 4513 sbthlem1 6985 subgintm 13134 neipsm 14106 neiuni 14113 |
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