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Theorem intssuni2 3903
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 3900 . 2  |-  ( A  =/=  (/)  ->  |^| A  C_  U. A )
2 uniss 3864 . 2  |-  ( A 
C_  B  ->  U. A  C_ 
U. B )
31, 2sylan9ssr 3206 1  |-  ( ( A  C_  B  /\  A  =/=  (/) )  ->  |^| A  C_ 
U. B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    =/= wne 2459    C_ wss 3165   (/)c0 3468   U.cuni 3843   |^|cint 3878
This theorem is referenced by:  rintn0  4008  fival  7182  mremre  13522  submre  13523  lssintcl  15737  iundifdifd  23175  iundifdif  23176  ismrcd1  26876
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-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-v 2803  df-dif 3168  df-in 3172  df-ss 3179  df-nul 3469  df-uni 3844  df-int 3879
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