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Theorem basis1 18514
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 18511 . . . 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 241 . . 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 3542 . . . . 5  |-  ( x  =  C  ->  (
x  i^i  y )  =  ( C  i^i  y ) )
43pweqd 3862 . . . . . . 7  |-  ( x  =  C  ->  ~P ( x  i^i  y
)  =  ~P ( C  i^i  y ) )
54ineq2d 3549 . . . . . 6  |-  ( x  =  C  ->  ( B  i^i  ~P ( x  i^i  y ) )  =  ( B  i^i  ~P ( C  i^i  y
) ) )
65unieqd 4098 . . . . 5  |-  ( x  =  C  ->  U. ( B  i^i  ~P ( x  i^i  y ) )  =  U. ( B  i^i  ~P ( C  i^i  y ) ) )
73, 6sseq12d 3382 . . . 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 3543 . . . . 5  |-  ( y  =  D  ->  ( C  i^i  y )  =  ( C  i^i  D
) )
98pweqd 3862 . . . . . . 7  |-  ( y  =  D  ->  ~P ( C  i^i  y
)  =  ~P ( C  i^i  D ) )
109ineq2d 3549 . . . . . 6  |-  ( y  =  D  ->  ( B  i^i  ~P ( C  i^i  y ) )  =  ( B  i^i  ~P ( C  i^i  D
) ) )
1110unieqd 4098 . . . . 5  |-  ( y  =  D  ->  U. ( B  i^i  ~P ( C  i^i  y ) )  =  U. ( B  i^i  ~P ( C  i^i  D ) ) )
128, 11sseq12d 3382 . . . 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 3076 . . 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 30 . 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 1180 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761   A.wral 2713    i^i cin 3324    C_ wss 3325   ~Pcpw 3857   U.cuni 4088   TopBasesctb 18461
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ral 2718  df-rex 2719  df-v 2972  df-in 3332  df-ss 3339  df-pw 3859  df-uni 4089  df-bases 18464
This theorem is referenced by: (None)
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