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Theorem basis1 12224
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 12221 . . . 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 175 . . 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 3270 . . . . 5  |-  ( x  =  C  ->  (
x  i^i  y )  =  ( C  i^i  y ) )
43pweqd 3515 . . . . . . 7  |-  ( x  =  C  ->  ~P ( x  i^i  y
)  =  ~P ( C  i^i  y ) )
54ineq2d 3277 . . . . . 6  |-  ( x  =  C  ->  ( B  i^i  ~P ( x  i^i  y ) )  =  ( B  i^i  ~P ( C  i^i  y
) ) )
65unieqd 3747 . . . . 5  |-  ( x  =  C  ->  U. ( B  i^i  ~P ( x  i^i  y ) )  =  U. ( B  i^i  ~P ( C  i^i  y ) ) )
73, 6sseq12d 3128 . . . 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 3271 . . . . 5  |-  ( y  =  D  ->  ( C  i^i  y )  =  ( C  i^i  D
) )
98pweqd 3515 . . . . . . 7  |-  ( y  =  D  ->  ~P ( C  i^i  y
)  =  ~P ( C  i^i  D ) )
109ineq2d 3277 . . . . . 6  |-  ( y  =  D  ->  ( B  i^i  ~P ( C  i^i  y ) )  =  ( B  i^i  ~P ( C  i^i  D
) ) )
1110unieqd 3747 . . . . 5  |-  ( y  =  D  ->  U. ( B  i^i  ~P ( C  i^i  y ) )  =  U. ( B  i^i  ~P ( C  i^i  D ) ) )
128, 11sseq12d 3128 . . . 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 2802 . . 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 29 . 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 1179 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 103    /\ w3a 962    = wceq 1331    e. wcel 1480   A.wral 2416    i^i cin 3070    C_ wss 3071   ~Pcpw 3510   U.cuni 3736   TopBasesctb 12219
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-in 3077  df-ss 3084  df-pw 3512  df-uni 3737  df-bases 12220
This theorem is referenced by: (None)
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