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

Theorem intss1 3846
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  |-  ( A  e.  B  ->  |^| B  C_  A )

Proof of Theorem intss1
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2733 . . . 4  |-  x  e. 
_V
21elint 3837 . . 3  |-  ( x  e.  |^| B  <->  A. y
( y  e.  B  ->  x  e.  y ) )
3 eleq1 2233 . . . . . 6  |-  ( y  =  A  ->  (
y  e.  B  <->  A  e.  B ) )
4 eleq2 2234 . . . . . 6  |-  ( y  =  A  ->  (
x  e.  y  <->  x  e.  A ) )
53, 4imbi12d 233 . . . . 5  |-  ( y  =  A  ->  (
( y  e.  B  ->  x  e.  y )  <-> 
( A  e.  B  ->  x  e.  A ) ) )
65spcgv 2817 . . . 4  |-  ( A  e.  B  ->  ( A. y ( y  e.  B  ->  x  e.  y )  ->  ( A  e.  B  ->  x  e.  A ) ) )
76pm2.43a 51 . . 3  |-  ( A  e.  B  ->  ( A. y ( y  e.  B  ->  x  e.  y )  ->  x  e.  A ) )
82, 7syl5bi 151 . 2  |-  ( A  e.  B  ->  (
x  e.  |^| B  ->  x  e.  A ) )
98ssrdv 3153 1  |-  ( A  e.  B  ->  |^| B  C_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1346    = wceq 1348    e. wcel 2141    C_ wss 3121   |^|cint 3831
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127  df-ss 3134  df-int 3832
This theorem is referenced by:  intminss  3856  intmin3  3858  intab  3860  int0el  3861  trintssm  4103  inteximm  4135  onnmin  4552  peano5  4582  peano5nnnn  7854  peano5nni  8881  dfuzi  9322  bj-intabssel  13824  bj-intabssel1  13825
  Copyright terms: Public domain W3C validator