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Metamath Proof Explorer

Theorem ween

Description: A set is numerable iff it can be well-ordered. (Contributed by Mario Carneiro, 5-Jan-2013)

Ref Expression
Assertion ween 
|- ( A e. dom card <-> E. r r We A )

Proof

Step Hyp Ref Expression
1 dfac8b 
 |-  ( A e. dom card -> E. r r We A )
2 weso 
 |-  ( r We A -> r Or A )
3 vex 
 |-  r e. _V
4 soex 
 |-  ( ( r Or A /\ r e. _V ) -> A e. _V )
5 2 3 4 sylancl 
 |-  ( r We A -> A e. _V )
6 5 exlimiv 
 |-  ( E. r r We A -> A e. _V )
7 unipw 
 |-  U. ~P A = A
8 weeq2 
 |-  ( U. ~P A = A -> ( r We U. ~P A <-> r We A ) )
9 7 8 ax-mp 
 |-  ( r We U. ~P A <-> r We A )
10 9 exbii 
 |-  ( E. r r We U. ~P A <-> E. r r We A )
11 10 biimpri 
 |-  ( E. r r We A -> E. r r We U. ~P A )
12 pwexg 
 |-  ( A e. _V -> ~P A e. _V )
13 dfac8c 
 |-  ( ~P A e. _V -> ( E. r r We U. ~P A -> E. f A. x e. ~P A ( x =/= (/) -> ( f ` x ) e. x ) ) )
14 12 13 syl 
 |-  ( A e. _V -> ( E. r r We U. ~P A -> E. f A. x e. ~P A ( x =/= (/) -> ( f ` x ) e. x ) ) )
15 dfac8a 
 |-  ( A e. _V -> ( E. f A. x e. ~P A ( x =/= (/) -> ( f ` x ) e. x ) -> A e. dom card ) )
16 14 15 syld 
 |-  ( A e. _V -> ( E. r r We U. ~P A -> A e. dom card ) )
17 6 11 16 sylc 
 |-  ( E. r r We A -> A e. dom card )
18 1 17 impbii 
 |-  ( A e. dom card <-> E. r r We A )