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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 3987 |
. . 3
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3 | eleq2 2148 |
. . . 4
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4 | 1 | snid 3452 |
. . . 4
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5 | 3, 4 | intmin3 3692 |
. . 3
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6 | 2, 5 | ax-mp 7 |
. 2
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7 | 1 | elintab 3676 |
. . . 4
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8 | id 19 |
. . . 4
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9 | 7, 8 | mpgbir 1385 |
. . 3
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10 | snssi 3558 |
. . 3
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11 | 9, 10 | ax-mp 7 |
. 2
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12 | 6, 11 | eqssi 3028 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1379 ax-7 1380 ax-gen 1381 ax-ie1 1425 ax-ie2 1426 ax-8 1438 ax-10 1439 ax-11 1440 ax-i12 1441 ax-bndl 1442 ax-4 1443 ax-14 1448 ax-17 1462 ax-i9 1466 ax-ial 1470 ax-i5r 1471 ax-ext 2067 ax-sep 3925 ax-pow 3977 |
This theorem depends on definitions: df-bi 115 df-tru 1290 df-nf 1393 df-sb 1690 df-clab 2072 df-cleq 2078 df-clel 2081 df-nfc 2214 df-v 2616 df-in 2992 df-ss 2999 df-pw 3411 df-sn 3431 df-int 3666 |
This theorem is referenced by: (None) |
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