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Metamath Proof Explorer

Theorem numufl

Description: Consequence of filssufilg : a set whose double powerset is well-orderable satisfies the ultrafilter lemma. (Contributed by Mario Carneiro, 26-Aug-2015)

Ref Expression
Assertion numufl 
|- ( ~P ~P X e. dom card -> X e. UFL )

Proof

Step Hyp Ref Expression
1 filssufilg 
 |-  ( ( f e. ( Fil ` X ) /\ ~P ~P X e. dom card ) -> E. g e. ( UFil ` X ) f C_ g )
2 1 ancoms 
 |-  ( ( ~P ~P X e. dom card /\ f e. ( Fil ` X ) ) -> E. g e. ( UFil ` X ) f C_ g )
3 2 ralrimiva 
 |-  ( ~P ~P X e. dom card -> A. f e. ( Fil ` X ) E. g e. ( UFil ` X ) f C_ g )
4 pwexr 
 |-  ( ~P ~P X e. dom card -> ~P X e. _V )
5 pwexb 
 |-  ( X e. _V <-> ~P X e. _V )
6 4 5 sylibr 
 |-  ( ~P ~P X e. dom card -> X e. _V )
7 isufl 
 |-  ( X e. _V -> ( X e. UFL <-> A. f e. ( Fil ` X ) E. g e. ( UFil ` X ) f C_ g ) )
8 6 7 syl 
 |-  ( ~P ~P X e. dom card -> ( X e. UFL <-> A. f e. ( Fil ` X ) E. g e. ( UFil ` X ) f C_ g ) )
9 3 8 mpbird 
 |-  ( ~P ~P X e. dom card -> X e. UFL )