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Mirrors > Home > MPE Home > Th. List > intid | Structured version Visualization version Unicode version |
Description: The intersection of all sets to which a set belongs is the singleton of that set. (Contributed by NM, 5-Jun-2009.) |
Ref | Expression |
---|---|
intid.1 |
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Ref | Expression |
---|---|
intid |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snex 4613 |
. . 3
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2 | eleq2 2518 |
. . . 4
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3 | intid.1 |
. . . . 5
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4 | 3 | snid 3963 |
. . . 4
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5 | 2, 4 | intmin3 4232 |
. . 3
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6 | 1, 5 | ax-mp 5 |
. 2
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7 | 3 | elintab 4214 |
. . . 4
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8 | id 22 |
. . . 4
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9 | 7, 8 | mpgbir 1676 |
. . 3
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10 | snssi 4084 |
. . 3
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11 | 9, 10 | ax-mp 5 |
. 2
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12 | 6, 11 | eqssi 3415 |
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-9 1899 ax-10 1918 ax-11 1923 ax-12 1936 ax-13 2091 ax-ext 2431 ax-sep 4496 ax-nul 4505 ax-pr 4611 |
This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-3an 988 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-ne 2623 df-v 3014 df-dif 3374 df-un 3376 df-in 3378 df-ss 3385 df-nul 3699 df-sn 3936 df-pr 3938 df-int 4204 |
This theorem is referenced by: (None) |
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