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Theorem ssex 4350
Description: The subset of a set is also a set. Exercise 3 of [TakeutiZaring] p. 22. This is one way to express the Axiom of Separation ax-sep 4333 (a.k.a. Subset Axiom). (Contributed by NM, 27-Apr-1994.)
Hypothesis
Ref Expression
ssex.1  |-  B  e. 
_V
Assertion
Ref Expression
ssex  |-  ( A 
C_  B  ->  A  e.  _V )

Proof of Theorem ssex
StepHypRef Expression
1 df-ss 3336 . 2  |-  ( A 
C_  B  <->  ( A  i^i  B )  =  A )
2 ssex.1 . . . 4  |-  B  e. 
_V
32inex2 4348 . . 3  |-  ( A  i^i  B )  e. 
_V
4 eleq1 2498 . . 3  |-  ( ( A  i^i  B )  =  A  ->  (
( A  i^i  B
)  e.  _V  <->  A  e.  _V ) )
53, 4mpbii 204 . 2  |-  ( ( A  i^i  B )  =  A  ->  A  e.  _V )
61, 5sylbi 189 1  |-  ( A 
C_  B  ->  A  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   _Vcvv 2958    i^i cin 3321    C_ wss 3322
This theorem is referenced by:  ssexi  4351  ssexg  4352  intex  4359  moabex  4425  ixpiunwdom  7562  omex  7601  tcss  7686  bndrank  7770  scottex  7814  aceq3lem  8006  cfslb  8151  dcomex  8332  axdc2lem  8333  grothpw  8706  grothpwex  8707  grothomex  8709  elnp  8869  hashfacen  11708  limsuple  12277  limsuplt  12278  limsupbnd1  12281  o1add2  12422  o1mul2  12423  o1sub2  12424  o1dif  12428  caucvgrlem  12471  fsumo1  12596  unbenlem  13281  ressbas2  13525  prdsval  13683  prdsbas  13685  rescbas  14034  reschom  14035  rescco  14037  acsmapd  14609  issubmnd  14729  eqgfval  14993  dfod2  15205  ablfac1b  15633  2basgen  17060  prdstopn  17665  ressust  18299  rectbntr0  18868  elcncf  18924  cncfcnvcn  18956  cmsss  19308  ovolctb2  19393  limcfval  19764  ellimc2  19769  limcflf  19773  limcres  19778  limcun  19787  dvfval  19789  lhop2  19904  taylfval  20280  ulmval  20301  xrlimcnp  20812  ressnm  24189  brsset  25739  mblfinlem3  26257  isfne4  26363  refssfne  26388  topjoin  26408  filbcmb  26456  cnpwstotbnd  26520  ismtyval  26523  isnumbasgrplem2  27260  islinds2  27274  bnj849  29370  ispsubsp  30616  ispsubclN  30808
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960  df-in 3329  df-ss 3336
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