Theorem pw2eng

Description: The power set of a set is equinumerous to set exponentiation with a base of ordinal 2o . (Contributed by FL, 22-Feb-2011) (Revised by Mario Carneiro, 1-Jul-2015)

	Ref	Expression
Assertion	pw2eng	|- ( A e. V -> ~P A ~~ ( 2o ^m A ) )

Proof

Step	Hyp	Ref	Expression
1		pwexg	|- ( A e. V -> ~P A e. _V )
2		ovexd	|- ( A e. V -> ( { (/) , { (/) } } ^m A ) e. _V )
3		id	|- ( A e. V -> A e. V )
4		0ex	|- (/) e. _V
5	4	a1i	|- ( A e. V -> (/) e. _V )
6		p0ex	|- { (/) } e. _V
7	6	a1i	|- ( A e. V -> { (/) } e. _V )
8		0nep0	|- (/) =/= { (/) }
9	8	a1i	|- ( A e. V -> (/) =/= { (/) } )
10		eqid	|- ( x e. ~P A |-> ( z e. A |-> if ( z e. x , { (/) } , (/) ) ) ) = ( x e. ~P A |-> ( z e. A |-> if ( z e. x , { (/) } , (/) ) ) )
11	3 5 7 9 10	pw2f1o	|- ( A e. V -> ( x e. ~P A |-> ( z e. A |-> if ( z e. x , { (/) } , (/) ) ) ) : ~P A -1-1-onto-> ( { (/) , { (/) } } ^m A ) )
12		f1oen2g	|- ( ( ~P A e. _V /\ ( { (/) , { (/) } } ^m A ) e. _V /\ ( x e. ~P A |-> ( z e. A |-> if ( z e. x , { (/) } , (/) ) ) ) : ~P A -1-1-onto-> ( { (/) , { (/) } } ^m A ) ) -> ~P A ~~ ( { (/) , { (/) } } ^m A ) )
13	1 2 11 12	syl3anc	|- ( A e. V -> ~P A ~~ ( { (/) , { (/) } } ^m A ) )
14		df2o2	|- 2o = { (/) , { (/) } }
15	14	oveq1i	|- ( 2o ^m A ) = ( { (/) , { (/) } } ^m A )
16	13 15	breqtrrdi	|- ( A e. V -> ~P A ~~ ( 2o ^m A ) )