Theorem intssuni2 4502

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

Step	Hyp	Ref	Expression
1		intssuni 4499	. 2 |- ( A =/= (/) -> |^| A C_ U. A )
2		uniss 4458	. 2 |- ( A C_ B -> U. A C_ U. B )
3	1, 2	sylan9ssr 3617	1 |- ( ( A C_ B /\ A =/= (/) ) -> |^| A C_ U. B )

Syntax hints:   -> wi 4   /\ wa 384   =/= wne 2794   C_ wss 3574   (/)c0 3915   U.cuni 4436   |^|cint 4475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916  df-uni 4437  df-int 4476

This theorem is referenced by:  rintn0  4619  fival  8318  mremre  16264  submre  16265  lssintcl  18964  iundifdifd  29380  iundifdif  29381  bj-ismoored2  33063  bj-ismooredr2  33065  ismrcd1  37261