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Theorem intssuni2 4238
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 4235 . 2  |-  ( A  =/=  (/)  ->  |^| A  C_  U. A )
2 uniss 4197 . 2  |-  ( A 
C_  B  ->  U. A  C_ 
U. B )
31, 2sylan9ssr 3444 1  |-  ( ( A  C_  B  /\  A  =/=  (/) )  ->  |^| A  C_ 
U. B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    =/= wne 2587    C_ wss 3402   (/)c0 3724   U.cuni 4176   |^|cint 4212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2016  ax-ext 2370
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-clab 2378  df-cleq 2384  df-clel 2387  df-nfc 2542  df-ne 2589  df-ral 2747  df-rex 2748  df-v 3049  df-dif 3405  df-in 3409  df-ss 3416  df-nul 3725  df-uni 4177  df-int 4213
This theorem is referenced by:  rintn0  4350  fival  7805  mremre  15030  submre  15031  lssintcl  17742  iundifdifd  27586  iundifdif  27587  ismrcd1  30832
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