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Theorem eqss 3062
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 1431 . 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 2094 . 2  |-  ( A  =  B  <->  A. x
( x  e.  A  <->  x  e.  B ) )
3 dfss2 3036 . . 3  |-  ( A 
C_  B  <->  A. x
( x  e.  A  ->  x  e.  B ) )
4 dfss2 3036 . . 3  |-  ( B 
C_  A  <->  A. x
( x  e.  B  ->  x  e.  A ) )
53, 4anbi12i 451 . 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 211 1  |-  ( A  =  B  <->  ( A  C_  B  /\  B  C_  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   A.wal 1297    = wceq 1299    e. wcel 1448    C_ wss 3021
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-11 1452  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-in 3027  df-ss 3034
This theorem is referenced by:  eqssi  3063  eqssd  3064  sseq1  3070  sseq2  3071  eqimss  3101  ssrabeq  3130  uneqin  3274  ss0b  3349  vss  3357  sssnm  3628  unidif  3715  ssunieq  3716  iuneq1  3773  iuneq2  3776  iunxdif2  3808  ssext  4081  pweqb  4083  eqopab2b  4139  pwunim  4146  soeq2  4176  iunpw  4339  ordunisuc2r  4368  tfi  4434  eqrel  4566  eqrelrel  4578  coeq1  4634  coeq2  4635  cnveq  4651  dmeq  4677  relssres  4793  xp11m  4913  xpcanm  4914  xpcan2m  4915  ssrnres  4917  fnres  5175  eqfnfv3  5452  fneqeql2  5461  fconst4m  5572  f1imaeq  5608  eqoprab2b  5761  fo1stresm  5990  fo2ndresm  5991  nnacan  6338  nnmcan  6345  ixpeq2  6536  sbthlemi3  6775  isprm2  11591  bastop1  12034  epttop  12041  opnneiid  12115  cnntr  12175  metequiv  12423  bj-sseq  12580  bdeq0  12646  bdvsn  12653  bdop  12654  bdeqsuc  12660  bj-om  12720
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