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Theorem ssex 3953
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 3934 (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 3001 . 2  |-  ( A 
C_  B  <->  ( A  i^i  B )  =  A )
2 ssex.1 . . . 4  |-  B  e. 
_V
32inex2 3951 . . 3  |-  ( A  i^i  B )  e. 
_V
4 eleq1 2147 . . 3  |-  ( ( A  i^i  B )  =  A  ->  (
( A  i^i  B
)  e.  _V  <->  A  e.  _V ) )
53, 4mpbii 146 . 2  |-  ( ( A  i^i  B )  =  A  ->  A  e.  _V )
61, 5sylbi 119 1  |-  ( A 
C_  B  ->  A  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1287    e. wcel 1436   _Vcvv 2615    i^i cin 2987    C_ wss 2988
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3934
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617  df-in 2994  df-ss 3001
This theorem is referenced by:  ssexi  3954  ssexg  3955  inteximm  3962  funimaexglem  5064  tfrexlem  6055  elinp  6980  negfi  10575
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