Theorem ssex 3455

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 3438 (a.k.a. Subset Axiom).

Hypothesis

Ref	Expression
ssex.1	|- B e. _V

Assertion

Ref	Expression
ssex	|- (A C_ B -> A e. _V)

Proof of Theorem ssex

Step	Hyp	Ref	Expression
1		df-ss 2605	. 2 |- (A C_ B <-> (A i^i B) = A)
2		ssex.1	. . . 4 |- B e. _V
3	2	inex2 3453	. . 3 |- (A i^i B) e. _V
4		eleq1 1957	. . 3 |- ((A i^i B) = A -> ((A i^i B) e. _V <-> A e. _V))
5	3, 4	mpbii 210	. 2 |- ((A i^i B) = A -> A e. _V)
6	1, 5	sylbi 216	1 |- (A C_ B -> A e. _V)

Syntax hints:   -> wi 3   = wceq 1298   e. wcel 1300  _Vcvv 2292   i^i cin 2592   C_ wss 2593

This theorem is referenced by:  ssexi 3456  ssexg 3457  intex 3465  elpm 5395  mapss 5405  ordtypelem4 5687  inf3lem7 5725  omex 5733  unbnn3 5746  bndrank 5793  scottex 5846  zorn2lem4 5953  ondomon 6008  elnp 6244  suplem2pr 6314  lbinfm 7257  elcncf 8527  unbenlem 8773  lpval 9019  lmclim 9241  vacnlem4 9670  grothpw 10134  grothpwex 10135  sh 10711  bnj879 12806  bnj880 12807  brsset 14069  supnuf 15041  supexr 15043  ordtypelem4OLD 15378  filclus 15605  filbcmb 15776  heiborlem1 15955  igenval 16209  iscsubsp 17209  ispsubsp 17226  ispsubcl 17347

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-in 2603  df-ss 2605

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