Theorem intssunim 3846

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 3495	. . . 4 |- ( ( E. x x e. A /\ A. x e. A y e. x ) -> E. x e. A y e. x )
2	1	ex 114	. . 3 |- ( E. x x e. A -> ( A. x e. A y e. x -> E. x e. A y e. x ) )
3		vex 2729	. . . 4 |- y e. _V
4	3	elint2 3831	. . 3 |- ( y e. |^| A <-> A. x e. A y e. x )
5		eluni2 3793	. . 3 |- ( y e. U. A <-> E. x e. A y e. x )
6	2, 4, 5	3imtr4g 204	. 2 |- ( E. x x e. A -> ( y e. |^| A -> y e. U. A ) )
7	6	ssrdv 3148	1 |- ( E. x x e. A -> |^| A C_ U. A )

Syntax hints:   -> wi 4   E.wex 1480   e. wcel 2136   A.wral 2444   E.wrex 2445   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:  intssuni2m  3848