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Mirrors > Home > NFE 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 3895 |
. . . . . 6
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2 | 1 | imbi1i 315 |
. . . . 5
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3 | 19.23v 1891 |
. . . . 5
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4 | 2, 3 | bitr4i 243 |
. . . 4
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5 | 4 | albii 1566 |
. . 3
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6 | alcom 1737 |
. . . 4
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7 | 19.21v 1890 |
. . . . . 6
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8 | impexp 433 |
. . . . . . . 8
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9 | bi2.04 350 |
. . . . . . . 8
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10 | 8, 9 | bitri 240 |
. . . . . . 7
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11 | 10 | albii 1566 |
. . . . . 6
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12 | dfss2 3263 |
. . . . . . 7
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13 | 12 | imbi2i 303 |
. . . . . 6
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14 | 7, 11, 13 | 3bitr4i 268 |
. . . . 5
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15 | 14 | albii 1566 |
. . . 4
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16 | 6, 15 | bitri 240 |
. . 3
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17 | 5, 16 | bitri 240 |
. 2
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18 | dfss2 3263 |
. 2
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19 | df-ral 2620 |
. 2
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20 | 17, 18, 19 | 3bitr4i 268 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2479 df-ral 2620 df-v 2862 df-nin 3212 df-compl 3213 df-in 3214 df-ss 3260 df-uni 3893 |
This theorem is referenced by: uniss2 3923 ssunieq 3925 sspwuni 4052 pwssb 4053 |
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