Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  intss1 Unicode version

Theorem intss1 3630
 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.)
Assertion
Ref Expression
intss1

Proof of Theorem intss1
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2560 . . . 4
21elint 3621 . . 3
3 eleq1 2100 . . . . . 6
4 eleq2 2101 . . . . . 6
53, 4imbi12d 223 . . . . 5
65spcgv 2640 . . . 4
76pm2.43a 45 . . 3
82, 7syl5bi 141 . 2
98ssrdv 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
 Copyright terms: Public domain W3C validator