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Theorem intssuni2 4017
Description: Subclass relationship for intersection and union. (Contributed by NM, 29-Jul-2006.)
Assertion
Ref Expression
intssuni2  |-  ( ( A  C_  B  /\  A  =/=  (/) )  ->  |^| A  C_ 
U. B )

Proof of Theorem intssuni2
StepHypRef Expression
1 intssuni 4014 . 2  |-  ( A  =/=  (/)  ->  |^| A  C_  U. A )
2 uniss 3978 . 2  |-  ( A 
C_  B  ->  U. A  C_ 
U. B )
31, 2sylan9ssr 3305 1  |-  ( ( A  C_  B  /\  A  =/=  (/) )  ->  |^| A  C_ 
U. B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    =/= wne 2550    C_ wss 3263   (/)c0 3571   U.cuni 3957   |^|cint 3992
This theorem is referenced by:  rintn0  4122  fival  7352  mremre  13756  submre  13757  lssintcl  15967  iundifdifd  23856  iundifdif  23857  ismrcd1  26443
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-v 2901  df-dif 3266  df-in 3270  df-ss 3277  df-nul 3572  df-uni 3958  df-int 3993
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