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Theorem ssex 3997
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 3978 (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 3026 . 2  |-  ( A 
C_  B  <->  ( A  i^i  B )  =  A )
2 ssex.1 . . . 4  |-  B  e. 
_V
32inex2 3995 . . 3  |-  ( A  i^i  B )  e. 
_V
4 eleq1 2157 . . 3  |-  ( ( A  i^i  B )  =  A  ->  (
( A  i^i  B
)  e.  _V  <->  A  e.  _V ) )
53, 4mpbii 147 . 2  |-  ( ( A  i^i  B )  =  A  ->  A  e.  _V )
61, 5sylbi 120 1  |-  ( A 
C_  B  ->  A  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1296    e. wcel 1445   _Vcvv 2633    i^i cin 3012    C_ wss 3013
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978
This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-v 2635  df-in 3019  df-ss 3026
This theorem is referenced by:  ssexi  3998  ssexg  3999  inteximm  4006  funimaexglem  5131  tfrexlem  6137  elinp  7130  negfi  10774  elcncf  12328
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