Theorem intssuni2m 3894

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 3892	. 2 |- ( E. x x e. A -> |^| A C_ U. A )
2		uniss 3856	. 2 |- ( A C_ B -> U. A C_ U. B )
3	1, 2	sylan9ssr 3193	1 |- ( ( A C_ B /\ E. x x e. A ) -> |^| A C_ U. B )

Syntax hints:   -> wi 4   /\ wa 104   E.wex 1503   e. wcel 2164   C_ wss 3153   U.cuni 3835   |^|cint 3870

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-in 3159  df-ss 3166  df-uni 3836  df-int 3871

This theorem is referenced by:  rintm  4005  onintonm  4549  fival  7029  fiuni  7037  lssintclm  13880