Theorem intssunim 3921

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 3555	. . . 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 2779	. . . 4 |- y e. _V
4	3	elint2 3906	. . 3 |- ( y e. |^| A <-> A. x e. A y e. x )
5		eluni2 3868	. . 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 3207	1 |- ( E. x x e. A -> |^| A C_ U. A )

Syntax hints:   -> wi 4   E.wex 1516   e. wcel 2178   A.wral 2486   E.wrex 2487   C_ wss 3174   U.cuni 3864   |^|cint 3899

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-in 3180  df-ss 3187  df-uni 3865  df-int 3900

This theorem is referenced by:  intssuni2m  3923  subgintm  13649