MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  basis1 Unicode version

Theorem basis1 16704
Description: Property of a basis. (Contributed by NM, 16-Jul-2006.)
Assertion
Ref Expression
basis1  |-  ( ( B  e.  TopBases  /\  C  e.  B  /\  D  e.  B )  ->  ( C  i^i  D )  C_  U. ( B  i^i  ~P ( C  i^i  D ) ) )

Proof of Theorem basis1
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isbasisg 16701 . . . 4  |-  ( B  e.  TopBases  ->  ( B  e.  TopBases  <->  A. x  e.  B  A. y  e.  B  (
x  i^i  y )  C_ 
U. ( B  i^i  ~P ( x  i^i  y
) ) ) )
21ibi 232 . . 3  |-  ( B  e.  TopBases  ->  A. x  e.  B  A. y  e.  B  ( x  i^i  y
)  C_  U. ( B  i^i  ~P ( x  i^i  y ) ) )
3 ineq1 3376 . . . . 5  |-  ( x  =  C  ->  (
x  i^i  y )  =  ( C  i^i  y ) )
43pweqd 3643 . . . . . . 7  |-  ( x  =  C  ->  ~P ( x  i^i  y
)  =  ~P ( C  i^i  y ) )
54ineq2d 3383 . . . . . 6  |-  ( x  =  C  ->  ( B  i^i  ~P ( x  i^i  y ) )  =  ( B  i^i  ~P ( C  i^i  y
) ) )
65unieqd 3854 . . . . 5  |-  ( x  =  C  ->  U. ( B  i^i  ~P ( x  i^i  y ) )  =  U. ( B  i^i  ~P ( C  i^i  y ) ) )
73, 6sseq12d 3220 . . . 4  |-  ( x  =  C  ->  (
( x  i^i  y
)  C_  U. ( B  i^i  ~P ( x  i^i  y ) )  <-> 
( C  i^i  y
)  C_  U. ( B  i^i  ~P ( C  i^i  y ) ) ) )
8 ineq2 3377 . . . . 5  |-  ( y  =  D  ->  ( C  i^i  y )  =  ( C  i^i  D
) )
98pweqd 3643 . . . . . . 7  |-  ( y  =  D  ->  ~P ( C  i^i  y
)  =  ~P ( C  i^i  D ) )
109ineq2d 3383 . . . . . 6  |-  ( y  =  D  ->  ( B  i^i  ~P ( C  i^i  y ) )  =  ( B  i^i  ~P ( C  i^i  D
) ) )
1110unieqd 3854 . . . . 5  |-  ( y  =  D  ->  U. ( B  i^i  ~P ( C  i^i  y ) )  =  U. ( B  i^i  ~P ( C  i^i  D ) ) )
128, 11sseq12d 3220 . . . 4  |-  ( y  =  D  ->  (
( C  i^i  y
)  C_  U. ( B  i^i  ~P ( C  i^i  y ) )  <-> 
( C  i^i  D
)  C_  U. ( B  i^i  ~P ( C  i^i  D ) ) ) )
137, 12rspc2v 2903 . . 3  |-  ( ( C  e.  B  /\  D  e.  B )  ->  ( A. x  e.  B  A. y  e.  B  ( x  i^i  y )  C_  U. ( B  i^i  ~P ( x  i^i  y ) )  ->  ( C  i^i  D )  C_  U. ( B  i^i  ~P ( C  i^i  D ) ) ) )
142, 13syl5com 26 . 2  |-  ( B  e.  TopBases  ->  ( ( C  e.  B  /\  D  e.  B )  ->  ( C  i^i  D )  C_  U. ( B  i^i  ~P ( C  i^i  D ) ) ) )
15143impib 1149 1  |-  ( ( B  e.  TopBases  /\  C  e.  B  /\  D  e.  B )  ->  ( C  i^i  D )  C_  U. ( B  i^i  ~P ( C  i^i  D ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556    i^i cin 3164    C_ wss 3165   ~Pcpw 3638   U.cuni 3843   TopBasesctb 16651
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-3an 936  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-rex 2562  df-v 2803  df-in 3172  df-ss 3179  df-pw 3640  df-uni 3844  df-bases 16654
  Copyright terms: Public domain W3C validator