Theorem intssuni2m 3848

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 3846	. 2 |- ( E. x x e. A -> |^| A C_ U. A )
2		uniss 3810	. 2 |- ( A C_ B -> U. A C_ U. B )
3	1, 2	sylan9ssr 3156	1 |- ( ( A C_ B /\ E. x x e. A ) -> |^| A C_ U. B )

Syntax hints:   -> wi 4   /\ wa 103   E.wex 1480   e. wcel 2136   C_ wss 3116   U.cuni 3789   |^|cint 3824

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-in 3122  df-ss 3129  df-uni 3790  df-int 3825

This theorem is referenced by:  rintm  3958  onintonm  4494  fival  6935  fiuni  6943