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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 4203 |
. . 3
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3 | eleq2 2253 |
. . . 4
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4 | 1 | snid 3638 |
. . . 4
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5 | 3, 4 | intmin3 3886 |
. . 3
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6 | 2, 5 | ax-mp 5 |
. 2
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7 | 1 | elintab 3870 |
. . . 4
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8 | id 19 |
. . . 4
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9 | 7, 8 | mpgbir 1464 |
. . 3
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10 | snssi 3751 |
. . 3
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11 | 9, 10 | ax-mp 5 |
. 2
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12 | 6, 11 | eqssi 3186 |
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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-int 3860 |
This theorem is referenced by: (None) |
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