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Theorem basis1 16688
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 16685 . . . 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 3363 . . . . 5  |-  ( x  =  C  ->  (
x  i^i  y )  =  ( C  i^i  y ) )
43pweqd 3630 . . . . . . 7  |-  ( x  =  C  ->  ~P ( x  i^i  y
)  =  ~P ( C  i^i  y ) )
54ineq2d 3370 . . . . . 6  |-  ( x  =  C  ->  ( B  i^i  ~P ( x  i^i  y ) )  =  ( B  i^i  ~P ( C  i^i  y
) ) )
65unieqd 3838 . . . . 5  |-  ( x  =  C  ->  U. ( B  i^i  ~P ( x  i^i  y ) )  =  U. ( B  i^i  ~P ( C  i^i  y ) ) )
73, 6sseq12d 3207 . . . 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 3364 . . . . 5  |-  ( y  =  D  ->  ( C  i^i  y )  =  ( C  i^i  D
) )
98pweqd 3630 . . . . . . 7  |-  ( y  =  D  ->  ~P ( C  i^i  y
)  =  ~P ( C  i^i  D ) )
109ineq2d 3370 . . . . . 6  |-  ( y  =  D  ->  ( B  i^i  ~P ( C  i^i  y ) )  =  ( B  i^i  ~P ( C  i^i  D
) ) )
1110unieqd 3838 . . . . 5  |-  ( y  =  D  ->  U. ( B  i^i  ~P ( C  i^i  y ) )  =  U. ( B  i^i  ~P ( C  i^i  D ) ) )
128, 11sseq12d 3207 . . . 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 2890 . . 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 1623    e. wcel 1684   A.wral 2543    i^i cin 3151    C_ wss 3152   ~Pcpw 3625   U.cuni 3827   TopBasesctb 16635
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-v 2790  df-in 3159  df-ss 3166  df-pw 3627  df-uni 3828  df-bases 16638
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