Theorem intssuni2m 3952

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 3950	. 2 |- ( E. x x e. A -> |^| A C_ U. A )
2		uniss 3914	. 2 |- ( A C_ B -> U. A C_ U. B )
3	1, 2	sylan9ssr 3241	1 |- ( ( A C_ B /\ E. x x e. A ) -> |^| A C_ U. B )

Syntax hints:   -> wi 4   /\ wa 104   E.wex 1540   e. wcel 2202   C_ wss 3200   U.cuni 3893   |^|cint 3928

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-in 3206  df-ss 3213  df-uni 3894  df-int 3929

This theorem is referenced by:  rintm  4063  onintonm  4615  fival  7168  fiuni  7176  lssintclm  14397