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Theorem intssuni2 4297
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 4294 . 2  |-  ( A  =/=  (/)  ->  |^| A  C_  U. A )
2 uniss 4255 . 2  |-  ( A 
C_  B  ->  U. A  C_ 
U. B )
31, 2sylan9ssr 3503 1  |-  ( ( A  C_  B  /\  A  =/=  (/) )  ->  |^| A  C_ 
U. B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    =/= wne 2638    C_ wss 3461   (/)c0 3770   U.cuni 4234   |^|cint 4271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-v 3097  df-dif 3464  df-in 3468  df-ss 3475  df-nul 3771  df-uni 4235  df-int 4272
This theorem is referenced by:  rintn0  4406  fival  7874  mremre  14983  submre  14984  lssintcl  17589  iundifdifd  27407  iundifdif  27408  ismrcd1  30606
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