Theorem intssunim 3881

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 3524	. . . 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 2755	. . . 4 |- y e. _V
4	3	elint2 3866	. . 3 |- ( y e. |^| A <-> A. x e. A y e. x )
5		eluni2 3828	. . 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 3176	1 |- ( E. x x e. A -> |^| A C_ U. A )

Syntax hints:   -> wi 4   E.wex 1503   e. wcel 2160   A.wral 2468   E.wrex 2469   C_ wss 3144   U.cuni 3824   |^|cint 3859

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 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-in 3150  df-ss 3157  df-uni 3825  df-int 3860

This theorem is referenced by:  intssuni2m  3883  subgintm  13154