MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ssiin Unicode version

Theorem ssiin 3968
Description: Subset theorem for an indexed intersection. (Contributed by NM, 15-Oct-2003.)
Assertion
Ref Expression
ssiin  |-  ( C 
C_  |^|_ x  e.  A  B 
<-> 
A. x  e.  A  C  C_  B )
Distinct variable group:    x, C
Allowed substitution hints:    A( x)    B( x)

Proof of Theorem ssiin
StepHypRef Expression
1 nfcv 2432 . 2  |-  F/_ x C
21ssiinf 3967 1  |-  ( C 
C_  |^|_ x  e.  A  B 
<-> 
A. x  e.  A  C  C_  B )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   A.wral 2556    C_ wss 3165   |^|_ciin 3922
This theorem is referenced by:  cflim2  7905  ptbasfi  17292  limciun  19260  inttop2  25618  clsint2  26350  fnemeet2  26419  dihglblem4  32109  dihglblem6  32152
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-v 2803  df-in 3172  df-ss 3179  df-iin 3924
  Copyright terms: Public domain W3C validator