ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  elin2 Unicode version

Theorem elin2 3315
Description: Membership in a class defined as an intersection. (Contributed by Stefan O'Rear, 29-Mar-2015.)
Hypothesis
Ref Expression
elin2.x  |-  X  =  ( B  i^i  C
)
Assertion
Ref Expression
elin2  |-  ( A  e.  X  <->  ( A  e.  B  /\  A  e.  C ) )

Proof of Theorem elin2
StepHypRef Expression
1 elin2.x . . 3  |-  X  =  ( B  i^i  C
)
21eleq2i 2237 . 2  |-  ( A  e.  X  <->  A  e.  ( B  i^i  C ) )
3 elin 3310 . 2  |-  ( A  e.  ( B  i^i  C )  <->  ( A  e.  B  /\  A  e.  C ) )
42, 3bitri 183 1  |-  ( A  e.  X  <->  ( A  e.  B  /\  A  e.  C ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    = wceq 1348    e. wcel 2141    i^i cin 3120
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
This theorem is referenced by:  elin3  3318  fnres  5314  funfvima  5727  lmres  13042
  Copyright terms: Public domain W3C validator