Theorem eqss 2960

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.

Step	Hyp	Ref	Expression
1		albiim 1376	. 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 2034	. 2 |- ( A = B <-> A. x ( x e. A <-> x e. B ) )
3		dfss2 2934	. . 3 |- ( A C_ B <-> A. x ( x e. A -> x e. B ) )
4		dfss2 2934	. . 3 |- ( B C_ A <-> A. x ( x e. B -> x e. A ) )
5	3, 4	anbi12i 433	. 2 |- ( ( A C_ B /\ B C_ A ) <-> ( A. x ( x e. A -> x e. B ) /\ A. x ( x e. B -> x e. A ) ) )
6	1, 2, 5	3bitr4i 201	1 |- ( A = B <-> ( A C_ B /\ B C_ A ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   A.wal 1241   = wceq 1243   e. wcel 1393   C_ wss 2917

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-in 2924  df-ss 2931

This theorem is referenced by:  eqssi  2961  eqssd  2962  sseq1  2966  sseq2  2967  eqimss  2997  ssrabeq  3026  dfpss3  3030  uneqin  3188  ss0b  3256  vss  3264  sssnm  3525  unidif  3612  ssunieq  3613  iuneq1  3670  iuneq2  3673  iunxdif2  3705  ssext  3957  pweqb  3959  eqopab2b  4016  pwunim  4023  soeq2  4053  iunpw  4211  ordunisuc2r  4240  tfi  4305  eqrel  4429  eqrelrel  4441  coeq1  4493  coeq2  4494  cnveq  4509  dmeq  4535  relssres  4648  xp11m  4759  xpcanm  4760  xpcan2m  4761  ssrnres  4763  fnres  5015  eqfnfv3  5267  fneqeql2  5276  fconst4m  5381  f1imaeq  5414  eqoprab2b  5563  fo1stresm  5788  fo2ndresm  5789  nnacan  6085  nnmcan  6092  bj-sseq  9931  bdeq0  9987  bdvsn  9994  bdop  9995  bdeqsuc  10001  bj-om  10061