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Theorem dfss3 2963
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 2962 . 2  |-  ( A 
C_  B  <->  A. x
( x  e.  A  ->  x  e.  B ) )
2 df-ral 2328 . 2  |-  ( A. x  e.  A  x  e.  B  <->  A. x ( x  e.  A  ->  x  e.  B ) )
31, 2bitr4i 180 1  |-  ( A 
C_  B  <->  A. x  e.  A  x  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 102   A.wal 1257    e. wcel 1409   A.wral 2323    C_ wss 2945
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-ral 2328  df-in 2952  df-ss 2959
This theorem is referenced by:  ssrab  3046  eqsnm  3554  uni0b  3633  uni0c  3634  ssint  3659  ssiinf  3734  sspwuni  3767  dftr3  3886  tfis  4334  rninxp  4792  fnres  5043  eqfnfv3  5295  funimass3  5311  ffvresb  5356  tfrlemibxssdm  5972  bdss  10371
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