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Theorem ssex 3922
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 3903 (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 2959 . 2  |-  ( A 
C_  B  <->  ( A  i^i  B )  =  A )
2 ssex.1 . . . 4  |-  B  e. 
_V
32inex2 3920 . . 3  |-  ( A  i^i  B )  e. 
_V
4 eleq1 2116 . . 3  |-  ( ( A  i^i  B )  =  A  ->  (
( A  i^i  B
)  e.  _V  <->  A  e.  _V ) )
53, 4mpbii 140 . 2  |-  ( ( A  i^i  B )  =  A  ->  A  e.  _V )
61, 5sylbi 118 1  |-  ( A 
C_  B  ->  A  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1259    e. wcel 1409   _Vcvv 2574    i^i cin 2944    C_ wss 2945
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-in 2952  df-ss 2959
This theorem is referenced by:  ssexi  3923  ssexg  3924  inteximm  3931  funimaexglem  5010  tfrexlem  5979  elinp  6630
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