Metamath Proof Explorer


Theorem wdomac

Description: When assuming AC, weak and usual dominance coincide. It is not known if this is an AC equivalent. (Contributed by Stefan O'Rear, 11-Feb-2015) (Revised by Mario Carneiro, 5-May-2015)

Ref Expression
Assertion wdomac
|- ( X ~<_* Y <-> X ~<_ Y )

Proof

Step Hyp Ref Expression
1 relwdom
 |-  Rel ~<_*
2 1 brrelex2i
 |-  ( X ~<_* Y -> Y e. _V )
3 reldom
 |-  Rel ~<_
4 3 brrelex2i
 |-  ( X ~<_ Y -> Y e. _V )
5 numth3
 |-  ( Y e. _V -> Y e. dom card )
6 wdomnumr
 |-  ( Y e. dom card -> ( X ~<_* Y <-> X ~<_ Y ) )
7 5 6 syl
 |-  ( Y e. _V -> ( X ~<_* Y <-> X ~<_ Y ) )
8 2 4 7 pm5.21nii
 |-  ( X ~<_* Y <-> X ~<_ Y )