Theorem intssunim 3637

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 3309	. . . 4 |- ( ( E. x x e. A /\ A. x e. A y e. x ) -> E. x e. A y e. x )
2	1	ex 108	. . 3 |- ( E. x x e. A -> ( A. x e. A y e. x -> E. x e. A y e. x ) )
3		vex 2560	. . . 4 |- y e. _V
4	3	elint2 3622	. . 3 |- ( y e. |^| A <-> A. x e. A y e. x )
5		eluni2 3584	. . 3 |- ( y e. U. A <-> E. x e. A y e. x )
6	2, 4, 5	3imtr4g 194	. 2 |- ( E. x x e. A -> ( y e. |^| A -> y e. U. A ) )
7	6	ssrdv 2951	1 |- ( E. x x e. A -> |^| A C_ U. A )

Syntax hints:   -> wi 4   E.wex 1381   e. wcel 1393   A.wral 2306   E.wrex 2307   C_ wss 2917   U.cuni 3580   |^|cint 3615

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-in 2924  df-ss 2931  df-uni 3581  df-int 3616

This theorem is referenced by:  intssuni2m  3639