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Mirrors > Home > ILE Home > Th. List > intid | 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 | intid.1 |
. . . 4
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2 | 1 | snex 4184 |
. . 3
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3 | eleq2 2241 |
. . . 4
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4 | 1 | snid 3623 |
. . . 4
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5 | 3, 4 | intmin3 3871 |
. . 3
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6 | 2, 5 | ax-mp 5 |
. 2
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7 | 1 | elintab 3855 |
. . . 4
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8 | id 19 |
. . . 4
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9 | 7, 8 | mpgbir 1453 |
. . 3
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10 | snssi 3736 |
. . 3
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11 | 9, 10 | ax-mp 5 |
. 2
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12 | 6, 11 | eqssi 3171 |
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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4120 ax-pow 4173 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-int 3845 |
This theorem is referenced by: (None) |
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