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Theorem ssex 4152
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 4133 (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 3154 . 2  |-  ( A 
C_  B  <->  ( A  i^i  B )  =  A )
2 ssex.1 . . . 4  |-  B  e. 
_V
32inex2 4150 . . 3  |-  ( A  i^i  B )  e. 
_V
4 eleq1 2250 . . 3  |-  ( ( A  i^i  B )  =  A  ->  (
( A  i^i  B
)  e.  _V  <->  A  e.  _V ) )
53, 4mpbii 148 . 2  |-  ( ( A  i^i  B )  =  A  ->  A  e.  _V )
61, 5sylbi 121 1  |-  ( A 
C_  B  ->  A  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1363    e. wcel 2158   _Vcvv 2749    i^i cin 3140    C_ wss 3141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169  ax-sep 4133
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-in 3147  df-ss 3154
This theorem is referenced by:  ssexi  4153  ssexg  4154  inteximm  4161  exmid1stab  4220  funimaexglem  5311  tfrexlem  6349  elinp  7487  suplocexprlem2b  7727  negfi  11250  ssomct  12460  ssnnctlemct  12461  nninfdc  12468  elcncf  14413  sbthom  15128
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