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Theorem dfss3 3055
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 3054 . 2  |-  ( A 
C_  B  <->  A. x
( x  e.  A  ->  x  e.  B ) )
2 df-ral 2396 . 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 1312    e. wcel 1463   A.wral 2391    C_ wss 3039
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 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-11 1467  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-ral 2396  df-in 3045  df-ss 3052
This theorem is referenced by:  ssrab  3143  eqsnm  3650  uni0b  3729  uni0c  3730  ssint  3755  ssiinf  3830  sspwuni  3865  dftr3  3998  tfis  4465  rninxp  4950  fnres  5207  eqfnfv3  5486  funimass3  5502  ffvresb  5549  tfrlemibxssdm  6190  tfr1onlembxssdm  6206  tfrcllembxssdm  6219  suplocsr  7581  isbasis2g  12107  tgval2  12115  eltg2b  12118  tgss2  12143  basgen2  12145  bastop1  12147  unicld  12180  neipsm  12218  ssidcn  12274  bdss  12873
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