ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dfss3 Unicode version

Theorem dfss3 3137
Description: Alternate definition of subclass relationship. (Contributed by NM, 14-Oct-1999.)
Assertion
Ref Expression
dfss3  |-  ( A 
C_  B  <->  A. x  e.  A  x  e.  B )
Distinct variable groups:    x, A    x, B

Proof of Theorem dfss3
StepHypRef Expression
1 dfss2 3136 . 2  |-  ( A 
C_  B  <->  A. x
( x  e.  A  ->  x  e.  B ) )
2 df-ral 2453 . 2  |-  ( A. x  e.  A  x  e.  B  <->  A. x ( x  e.  A  ->  x  e.  B ) )
31, 2bitr4i 186 1  |-  ( A 
C_  B  <->  A. x  e.  A  x  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   A.wal 1346    e. wcel 2141   A.wral 2448    C_ wss 3121
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-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  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-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-ral 2453  df-in 3127  df-ss 3134
This theorem is referenced by:  ssrab  3225  eqsnm  3742  uni0b  3821  uni0c  3822  ssint  3847  ssiinf  3922  sspwuni  3957  dftr3  4091  tfis  4567  rninxp  5054  fnres  5314  eqfnfv3  5595  funimass3  5612  ffvresb  5659  tfrlemibxssdm  6306  tfr1onlembxssdm  6322  tfrcllembxssdm  6335  exmidontriimlem3  7200  suplocsr  7771  isbasis2g  12837  tgval2  12845  eltg2b  12848  tgss2  12873  basgen2  12875  bastop1  12877  unicld  12910  neipsm  12948  ssidcn  13004  bdss  13899
  Copyright terms: Public domain W3C validator