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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 |
Ref | Expression |
---|---|
intid |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | intid.1 | . . . 4 | |
2 | 1 | snex 4079 | . . 3 |
3 | eleq2 2181 | . . . 4 | |
4 | 1 | snid 3526 | . . . 4 |
5 | 3, 4 | intmin3 3768 | . . 3 |
6 | 2, 5 | ax-mp 5 | . 2 |
7 | 1 | elintab 3752 | . . . 4 |
8 | id 19 | . . . 4 | |
9 | 7, 8 | mpgbir 1414 | . . 3 |
10 | snssi 3634 | . . 3 | |
11 | 9, 10 | ax-mp 5 | . 2 |
12 | 6, 11 | eqssi 3083 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1316 wcel 1465 cab 2103 cvv 2660 wss 3041 csn 3497 cint 3741 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-int 3742 |
This theorem is referenced by: (None) |
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