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Metamath Proof Explorer

Theorem ficard

Description: A set is finite iff its cardinal is a natural number. (Contributed by Jeff Madsen, 2-Sep-2009)

Ref Expression
Assertion ficard 
|- ( A e. V -> ( A e. Fin <-> ( card ` A ) e. _om ) )

Proof

Step Hyp Ref Expression
1 isfi 
 |-  ( A e. Fin <-> E. x e. _om A ~~ x )
2 carden 
 |-  ( ( A e. V /\ x e. _om ) -> ( ( card ` A ) = ( card ` x ) <-> A ~~ x ) )
3 cardnn 
 |-  ( x e. _om -> ( card ` x ) = x )
4 eqtr 
 |-  ( ( ( card ` A ) = ( card ` x ) /\ ( card ` x ) = x ) -> ( card ` A ) = x )
5 4 expcom 
 |-  ( ( card ` x ) = x -> ( ( card ` A ) = ( card ` x ) -> ( card ` A ) = x ) )
6 3 5 syl 
 |-  ( x e. _om -> ( ( card ` A ) = ( card ` x ) -> ( card ` A ) = x ) )
7 eleq1a 
 |-  ( x e. _om -> ( ( card ` A ) = x -> ( card ` A ) e. _om ) )
8 6 7 syld 
 |-  ( x e. _om -> ( ( card ` A ) = ( card ` x ) -> ( card ` A ) e. _om ) )
9 8 adantl 
 |-  ( ( A e. V /\ x e. _om ) -> ( ( card ` A ) = ( card ` x ) -> ( card ` A ) e. _om ) )
10 2 9 sylbird 
 |-  ( ( A e. V /\ x e. _om ) -> ( A ~~ x -> ( card ` A ) e. _om ) )
11 10 rexlimdva 
 |-  ( A e. V -> ( E. x e. _om A ~~ x -> ( card ` A ) e. _om ) )
12 1 11 biimtrid 
 |-  ( A e. V -> ( A e. Fin -> ( card ` A ) e. _om ) )
13 cardnn 
 |-  ( ( card ` A ) e. _om -> ( card ` ( card ` A ) ) = ( card ` A ) )
14 13 eqcomd 
 |-  ( ( card ` A ) e. _om -> ( card ` A ) = ( card ` ( card ` A ) ) )
15 14 adantl 
 |-  ( ( A e. V /\ ( card ` A ) e. _om ) -> ( card ` A ) = ( card ` ( card ` A ) ) )
16 carden 
 |-  ( ( A e. V /\ ( card ` A ) e. _om ) -> ( ( card ` A ) = ( card ` ( card ` A ) ) <-> A ~~ ( card ` A ) ) )
17 15 16 mpbid 
 |-  ( ( A e. V /\ ( card ` A ) e. _om ) -> A ~~ ( card ` A ) )
18 17 ex 
 |-  ( A e. V -> ( ( card ` A ) e. _om -> A ~~ ( card ` A ) ) )
19 18 ancld 
 |-  ( A e. V -> ( ( card ` A ) e. _om -> ( ( card ` A ) e. _om /\ A ~~ ( card ` A ) ) ) )
20 breq2 
 |-  ( x = ( card ` A ) -> ( A ~~ x <-> A ~~ ( card ` A ) ) )
21 20 rspcev 
 |-  ( ( ( card ` A ) e. _om /\ A ~~ ( card ` A ) ) -> E. x e. _om A ~~ x )
22 21 1 sylibr 
 |-  ( ( ( card ` A ) e. _om /\ A ~~ ( card ` A ) ) -> A e. Fin )
23 19 22 syl6 
 |-  ( A e. V -> ( ( card ` A ) e. _om -> A e. Fin ) )
24 12 23 impbid 
 |-  ( A e. V -> ( A e. Fin <-> ( card ` A ) e. _om ) )