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Metamath Proof Explorer

Theorem axccd2

Description: An alternative version of the axiom of countable choice. (Contributed by Glauco Siliprandi, 26-Jun-2021)

Ref Expression
Hypotheses axccd2.1 
|- ( ph -> A ~<_ _om )
axccd2.2 
|- ( ( ph /\ x e. A ) -> x =/= (/) )
Assertion axccd2 
|- ( ph -> E. f A. x e. A ( f ` x ) e. x )

Proof

Step Hyp Ref Expression
1 axccd2.1 
 |-  ( ph -> A ~<_ _om )
2 axccd2.2 
 |-  ( ( ph /\ x e. A ) -> x =/= (/) )
3 isfinite2 
 |-  ( A ~< _om -> A e. Fin )
4 3 adantl 
 |-  ( ( ph /\ A ~< _om ) -> A e. Fin )
5 simpr 
 |-  ( ( ( ph /\ A ~< _om ) /\ x e. A ) -> x e. A )
6 2 adantlr 
 |-  ( ( ( ph /\ A ~< _om ) /\ x e. A ) -> x =/= (/) )
7 4 5 6 choicefi 
 |-  ( ( ph /\ A ~< _om ) -> E. f ( f Fn A /\ A. x e. A ( f ` x ) e. x ) )
8 simpr 
 |-  ( ( f Fn A /\ A. x e. A ( f ` x ) e. x ) -> A. x e. A ( f ` x ) e. x )
9 8 a1i 
 |-  ( ( ph /\ A ~< _om ) -> ( ( f Fn A /\ A. x e. A ( f ` x ) e. x ) -> A. x e. A ( f ` x ) e. x ) )
10 9 eximdv 
 |-  ( ( ph /\ A ~< _om ) -> ( E. f ( f Fn A /\ A. x e. A ( f ` x ) e. x ) -> E. f A. x e. A ( f ` x ) e. x ) )
11 7 10 mpd 
 |-  ( ( ph /\ A ~< _om ) -> E. f A. x e. A ( f ` x ) e. x )
12 1 anim1i 
 |-  ( ( ph /\ -. A ~< _om ) -> ( A ~<_ _om /\ -. A ~< _om ) )
13 bren2 
 |-  ( A ~~ _om <-> ( A ~<_ _om /\ -. A ~< _om ) )
14 12 13 sylibr 
 |-  ( ( ph /\ -. A ~< _om ) -> A ~~ _om )
15 simpr 
 |-  ( ( ph /\ A ~~ _om ) -> A ~~ _om )
16 2 adantlr 
 |-  ( ( ( ph /\ A ~~ _om ) /\ x e. A ) -> x =/= (/) )
17 15 16 axccd 
 |-  ( ( ph /\ A ~~ _om ) -> E. f A. x e. A ( f ` x ) e. x )
18 14 17 syldan 
 |-  ( ( ph /\ -. A ~< _om ) -> E. f A. x e. A ( f ` x ) e. x )
19 11 18 pm2.61dan 
 |-  ( ph -> E. f A. x e. A ( f ` x ) e. x )