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Theorem intssuni2 4251
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 4248 . 2  |-  ( A  =/=  (/)  ->  |^| A  C_  U. A )
2 uniss 4211 . 2  |-  ( A 
C_  B  ->  U. A  C_ 
U. B )
31, 2sylan9ssr 3432 1  |-  ( ( A  C_  B  /\  A  =/=  (/) )  ->  |^| A  C_ 
U. B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    =/= wne 2641    C_ wss 3390   (/)c0 3722   U.cuni 4190   |^|cint 4226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-v 3033  df-dif 3393  df-in 3397  df-ss 3404  df-nul 3723  df-uni 4191  df-int 4227
This theorem is referenced by:  rintn0  4365  fival  7944  mremre  15588  submre  15589  lssintcl  18265  iundifdifd  28254  iundifdif  28255  ismrcd1  35611
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