| Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 2775 |
. . . 4
| |
| 2 | 1 | elint 3891 |
. . 3
|
| 3 | eleq1 2268 |
. . . . . 6
| |
| 4 | eleq2 2269 |
. . . . . 6
| |
| 5 | 3, 4 | imbi12d 234 |
. . . . 5
|
| 6 | 5 | spcgv 2860 |
. . . 4
|
| 7 | 6 | pm2.43a 51 |
. . 3
|
| 8 | 2, 7 | biimtrid 152 |
. 2
|
| 9 | 8 | ssrdv 3199 |
1
|
| 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 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-ext 2187 |
| This theorem depends on definitions: df-bi 117 df-tru 1376 df-nf 1484 df-sb 1786 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-v 2774 df-in 3172 df-ss 3179 df-int 3886 |
| This theorem is referenced by: intminss 3910 intmin3 3912 intab 3914 int0el 3915 trintssm 4158 inteximm 4193 onnmin 4616 peano5 4646 peano5nnnn 8005 peano5nni 9039 dfuzi 9483 bj-intabssel 15725 bj-intabssel1 15726 |
| Copyright terms: Public domain | W3C validator |