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Theorem eqss 3170
Description: The subclass relationship is antisymmetric. Compare Theorem 4 of [Suppes] p. 22. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
eqss  |-  ( A  =  B  <->  ( A  C_  B  /\  B  C_  A ) )

Proof of Theorem eqss
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 albiim 1487 . 2  |-  ( A. x ( x  e.  A  <->  x  e.  B
)  <->  ( A. x
( x  e.  A  ->  x  e.  B )  /\  A. x ( x  e.  B  ->  x  e.  A )
) )
2 dfcleq 2171 . 2  |-  ( A  =  B  <->  A. x
( x  e.  A  <->  x  e.  B ) )
3 dfss2 3144 . . 3  |-  ( A 
C_  B  <->  A. x
( x  e.  A  ->  x  e.  B ) )
4 dfss2 3144 . . 3  |-  ( B 
C_  A  <->  A. x
( x  e.  B  ->  x  e.  A ) )
53, 4anbi12i 460 . 2  |-  ( ( A  C_  B  /\  B  C_  A )  <->  ( A. x ( x  e.  A  ->  x  e.  B )  /\  A. x ( x  e.  B  ->  x  e.  A ) ) )
61, 2, 53bitr4i 212 1  |-  ( A  =  B  <->  ( A  C_  B  /\  B  C_  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105   A.wal 1351    = wceq 1353    e. wcel 2148    C_ wss 3129
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-in 3135  df-ss 3142
This theorem is referenced by:  eqssi  3171  eqssd  3172  sseq1  3178  sseq2  3179  eqimss  3209  ssrabeq  3242  uneqin  3386  ss0b  3462  vss  3470  sssnm  3754  unidif  3841  ssunieq  3842  iuneq1  3899  iuneq2  3902  iunxdif2  3935  ssext  4221  pweqb  4223  eqopab2b  4279  pwunim  4286  soeq2  4316  iunpw  4480  ordunisuc2r  4513  tfi  4581  eqrel  4715  eqrelrel  4727  coeq1  4784  coeq2  4785  cnveq  4801  dmeq  4827  relssres  4945  xp11m  5067  xpcanm  5068  xpcan2m  5069  ssrnres  5071  fnres  5332  eqfnfv3  5615  fneqeql2  5625  fconst4m  5736  f1imaeq  5775  eqoprab2b  5932  fo1stresm  6161  fo2ndresm  6162  nnacan  6512  nnmcan  6519  ixpeq2  6711  sbthlemi3  6957  isprm2  12116  bastop1  13553  epttop  13560  opnneiid  13634  cnntr  13695  metequiv  13965  bj-sseq  14514  bdeq0  14589  bdvsn  14596  bdop  14597  bdeqsuc  14603  bj-om  14659
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