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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 2715 | . . . 4 | |
2 | 1 | elint 3813 | . . 3 |
3 | eleq1 2220 | . . . . . 6 | |
4 | eleq2 2221 | . . . . . 6 | |
5 | 3, 4 | imbi12d 233 | . . . . 5 |
6 | 5 | spcgv 2799 | . . . 4 |
7 | 6 | pm2.43a 51 | . . 3 |
8 | 2, 7 | syl5bi 151 | . 2 |
9 | 8 | ssrdv 3134 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wal 1333 wceq 1335 wcel 2128 wss 3102 cint 3807 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-v 2714 df-in 3108 df-ss 3115 df-int 3808 |
This theorem is referenced by: intminss 3832 intmin3 3834 intab 3836 int0el 3837 trintssm 4078 inteximm 4110 onnmin 4527 peano5 4557 peano5nnnn 7812 peano5nni 8836 dfuzi 9274 bj-intabssel 13374 bj-intabssel1 13375 |
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