Theorem intssuni2m 3842

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 3840	. 2 |- ( E. x x e. A -> |^| A C_ U. A )
2		uniss 3804	. 2 |- ( A C_ B -> U. A C_ U. B )
3	1, 2	sylan9ssr 3151	1 |- ( ( A C_ B /\ E. x x e. A ) -> |^| A C_ U. B )

Syntax hints:   -> wi 4   /\ wa 103   E.wex 1479   e. wcel 2135   C_ wss 3111   U.cuni 3783   |^|cint 3818

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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-v 2723  df-in 3117  df-ss 3124  df-uni 3784  df-int 3819

This theorem is referenced by:  rintm  3952  onintonm  4488  fival  6926  fiuni  6934