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Mirrors > Home > MPE Home > Th. List > unissb | Structured version Visualization version 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 4170 |
. . . . . 6
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2 | 1 | imbi1i 331 |
. . . . 5
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3 | 19.23v 1821 |
. . . . 5
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4 | 2, 3 | bitr4i 260 |
. . . 4
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5 | 4 | albii 1694 |
. . 3
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6 | alcom 1926 |
. . . 4
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7 | 19.21v 1789 |
. . . . . 6
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8 | impexp 452 |
. . . . . . . 8
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9 | bi2.04 367 |
. . . . . . . 8
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10 | 8, 9 | bitri 257 |
. . . . . . 7
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11 | 10 | albii 1694 |
. . . . . 6
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12 | dfss2 3388 |
. . . . . . 7
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13 | 12 | imbi2i 318 |
. . . . . 6
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14 | 7, 11, 13 | 3bitr4i 285 |
. . . . 5
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15 | 14 | albii 1694 |
. . . 4
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16 | 6, 15 | bitri 257 |
. . 3
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17 | 5, 16 | bitri 257 |
. 2
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18 | dfss2 3388 |
. 2
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19 | df-ral 2741 |
. 2
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20 | 17, 18, 19 | 3bitr4i 285 |
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 1672 ax-4 1685 ax-5 1761 ax-6 1808 ax-7 1854 ax-10 1918 ax-11 1923 ax-12 1936 ax-13 2091 ax-ext 2431 |
This theorem depends on definitions: df-bi 190 df-an 377 df-tru 1450 df-ex 1667 df-nf 1671 df-sb 1801 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2581 df-ral 2741 df-v 3014 df-in 3378 df-ss 3385 df-uni 4168 |
This theorem is referenced by: uniss2 4199 ssunieq 4201 sspwuni 4338 pwssb 4339 ordunisssuc 5503 sorpssuni 6567 bm2.5ii 6620 sbthlem1 7668 ordunifi 7807 isfinite2 7815 cflim2 8679 fin23lem16 8751 fin23lem29 8757 fin1a2lem11 8826 fin1a2lem13 8828 itunitc 8837 zorng 8920 wuncval2 9158 suplem1pr 9463 suplem2pr 9464 mrcuni 15537 ipodrsfi 16419 mrelatlub 16442 subgint 16851 efgval 17377 toponmre 20119 neips 20139 neiuni 20148 alexsubALTlem2 21073 alexsubALTlem3 21074 tgpconcompeqg 21136 unidmvol 22505 tglnunirn 24604 uniinn0 28173 locfinreflem 28673 sxbrsigalem0 29098 dya2iocuni 29110 dya2iocucvr 29111 carsguni 29145 topjoin 31026 fnejoin1 31029 fnejoin2 31030 ovoliunnfl 31983 voliunnfl 31985 volsupnfl 31986 intidl 32263 unichnidl 32265 salexct 38249 |
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