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Mirrors > Home > ILE Home > Th. List > intss1 | Unicode version |
Description: An element of a class includes the intersection of the class. Exercise 4 of [TakeutiZaring] p. 44 (with correction), generalized to classes. (Contributed by NM, 18-Nov-1995.) |
Ref | Expression |
---|---|
intss1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2560 | . . . 4 | |
2 | 1 | elint 3621 | . . 3 |
3 | eleq1 2100 | . . . . . 6 | |
4 | eleq2 2101 | . . . . . 6 | |
5 | 3, 4 | imbi12d 223 | . . . . 5 |
6 | 5 | spcgv 2640 | . . . 4 |
7 | 6 | pm2.43a 45 | . . 3 |
8 | 2, 7 | syl5bi 141 | . 2 |
9 | 8 | ssrdv 2951 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wal 1241 wceq 1243 wcel 1393 wss 2917 cint 3615 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-in 2924 df-ss 2931 df-int 3616 |
This theorem is referenced by: intminss 3640 intmin3 3642 intab 3644 int0el 3645 trint0m 3871 inteximm 3903 onnmin 4292 peano5 4321 peano5nnnn 6966 peano5nni 7917 dfuzi 8348 bj-intabssel 9928 bj-intabssel1 9929 peano5setOLD 10065 |
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