Theorem pwun

Description: The power class of the union of two classes equals the union of their power classes, iff one class is a subclass of the other. Part of Exercise 7(b) of Enderton p. 28. (Contributed by NM, 23-Nov-2003)

	Ref	Expression
Assertion	pwun	|- ( ( A C_ B \/ B C_ A ) <-> ~P ( A u. B ) = ( ~P A u. ~P B ) )

Proof

Step	Hyp	Ref	Expression
1		pwunss	|- ( ~P A u. ~P B ) C_ ~P ( A u. B )
2	1	biantru	|- ( ~P ( A u. B ) C_ ( ~P A u. ~P B ) <-> ( ~P ( A u. B ) C_ ( ~P A u. ~P B ) /\ ( ~P A u. ~P B ) C_ ~P ( A u. B ) ) )
3		pwssun	|- ( ( A C_ B \/ B C_ A ) <-> ~P ( A u. B ) C_ ( ~P A u. ~P B ) )
4		eqss	|- ( ~P ( A u. B ) = ( ~P A u. ~P B ) <-> ( ~P ( A u. B ) C_ ( ~P A u. ~P B ) /\ ( ~P A u. ~P B ) C_ ~P ( A u. B ) ) )
5	2 3 4	3bitr4i	|- ( ( A C_ B \/ B C_ A ) <-> ~P ( A u. B ) = ( ~P A u. ~P B ) )