Theorem intssuni2m 3855

Description: Subclass relationship for intersection and union. (Contributed by Jim Kingdon, 14-Aug-2018.)

Assertion

Ref	Expression
intssuni2m	|- ( ( A C_ B /\ E. x x e. A ) -> |^| A C_ U. B )

Distinct variable group:   x, A

Allowed substitution hint:   B( x)

Proof of Theorem intssuni2m

Step	Hyp	Ref	Expression
1		intssunim 3853	. 2 |- ( E. x x e. A -> |^| A C_ U. A )
2		uniss 3817	. 2 |- ( A C_ B -> U. A C_ U. B )
3	1, 2	sylan9ssr 3161	1 |- ( ( A C_ B /\ E. x x e. A ) -> |^| A C_ U. B )

Syntax hints:   -> wi 4   /\ wa 103   E.wex 1485   e. wcel 2141   C_ wss 3121   U.cuni 3796   |^|cint 3831

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-in 3127  df-ss 3134  df-uni 3797  df-int 3832

This theorem is referenced by:  rintm  3965  onintonm  4501  fival  6947  fiuni  6955