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Theorem eqss 3116
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 1464 . 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 2134 . 2  |-  ( A  =  B  <->  A. x
( x  e.  A  <->  x  e.  B ) )
3 dfss2 3090 . . 3  |-  ( A 
C_  B  <->  A. x
( x  e.  A  ->  x  e.  B ) )
4 dfss2 3090 . . 3  |-  ( B 
C_  A  <->  A. x
( x  e.  B  ->  x  e.  A ) )
53, 4anbi12i 456 . 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 1330    = wceq 1332    e. wcel 1481    C_ wss 3075
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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-in 3081  df-ss 3088
This theorem is referenced by:  eqssi  3117  eqssd  3118  sseq1  3124  sseq2  3125  eqimss  3155  ssrabeq  3187  uneqin  3331  ss0b  3406  vss  3414  sssnm  3688  unidif  3775  ssunieq  3776  iuneq1  3833  iuneq2  3836  iunxdif2  3868  ssext  4150  pweqb  4152  eqopab2b  4208  pwunim  4215  soeq2  4245  iunpw  4408  ordunisuc2r  4437  tfi  4503  eqrel  4635  eqrelrel  4647  coeq1  4703  coeq2  4704  cnveq  4720  dmeq  4746  relssres  4864  xp11m  4984  xpcanm  4985  xpcan2m  4986  ssrnres  4988  fnres  5246  eqfnfv3  5527  fneqeql2  5536  fconst4m  5647  f1imaeq  5683  eqoprab2b  5836  fo1stresm  6066  fo2ndresm  6067  nnacan  6415  nnmcan  6422  ixpeq2  6613  sbthlemi3  6854  isprm2  11832  bastop1  12289  epttop  12296  opnneiid  12370  cnntr  12431  metequiv  12701  bj-sseq  13168  bdeq0  13234  bdvsn  13241  bdop  13242  bdeqsuc  13248  bj-om  13304
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