Theorem intssunim 3868

Description: The intersection of an inhabited set is a subclass of its union. (Contributed by NM, 29-Jul-2006.)

Assertion

Ref	Expression
intssunim	|- ( E. x x e. A -> |^| A C_ U. A )

Distinct variable group:   x, A

Proof of Theorem intssunim

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		r19.2m 3511	. . . 4 |- ( ( E. x x e. A /\ A. x e. A y e. x ) -> E. x e. A y e. x )
2	1	ex 115	. . 3 |- ( E. x x e. A -> ( A. x e. A y e. x -> E. x e. A y e. x ) )
3		vex 2742	. . . 4 |- y e. _V
4	3	elint2 3853	. . 3 |- ( y e. |^| A <-> A. x e. A y e. x )
5		eluni2 3815	. . 3 |- ( y e. U. A <-> E. x e. A y e. x )
6	2, 4, 5	3imtr4g 205	. 2 |- ( E. x x e. A -> ( y e. |^| A -> y e. U. A ) )
7	6	ssrdv 3163	1 |- ( E. x x e. A -> |^| A C_ U. A )

Syntax hints:   -> wi 4   E.wex 1492   e. wcel 2148   A.wral 2455   E.wrex 2456   C_ wss 3131   U.cuni 3811   |^|cint 3846

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-in 3137  df-ss 3144  df-uni 3812  df-int 3847

This theorem is referenced by:  intssuni2m  3870  subgintm  13064