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Theorem eqss 3185
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 1498 . 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 2183 . 2  |-  ( A  =  B  <->  A. x
( x  e.  A  <->  x  e.  B ) )
3 dfss2 3159 . . 3  |-  ( A 
C_  B  <->  A. x
( x  e.  A  ->  x  e.  B ) )
4 dfss2 3159 . . 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 1362    = wceq 1364    e. wcel 2160    C_ wss 3144
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-in 3150  df-ss 3157
This theorem is referenced by:  eqssi  3186  eqssd  3187  sseq1  3193  sseq2  3194  eqimss  3224  ssrabeq  3257  uneqin  3401  ss0b  3477  vss  3485  sssnm  3769  unidif  3856  ssunieq  3857  iuneq1  3914  iuneq2  3917  iunxdif2  3950  ssext  4236  pweqb  4238  eqopab2b  4294  pwunim  4301  soeq2  4331  iunpw  4495  ordunisuc2r  4528  tfi  4596  eqrel  4730  eqrelrel  4742  coeq1  4799  coeq2  4800  cnveq  4816  dmeq  4842  relssres  4960  xp11m  5082  xpcanm  5083  xpcan2m  5084  ssrnres  5086  fnres  5347  eqfnfv3  5631  fneqeql2  5641  fconst4m  5752  f1imaeq  5792  eqoprab2b  5949  fo1stresm  6180  fo2ndresm  6181  nnacan  6531  nnmcan  6538  ixpeq2  6730  sbthlemi3  6976  isprm2  12135  lssle0  13649  bastop1  13980  epttop  13987  opnneiid  14061  cnntr  14122  metequiv  14392  bj-sseq  14941  bdeq0  15016  bdvsn  15023  bdop  15024  bdeqsuc  15030  bj-om  15086
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