let W be Universe; :: thesis: ( omega in W implies ex a being Ordinal of W ex M being non empty set st

( a is_cofinal_with omega & M = Rank a & M <==> W ) )

assume omega in W ; :: thesis: ex a being Ordinal of W ex M being non empty set st

( a is_cofinal_with omega & M = Rank a & M <==> W )

then consider b being Ordinal of W, M being non empty set such that

A1: ( b is_cofinal_with omega & M = Rank b & M is_elementary_subsystem_of W ) by Th35;

take b ; :: thesis: ex M being non empty set st

( b is_cofinal_with omega & M = Rank b & M <==> W )

take M ; :: thesis: ( b is_cofinal_with omega & M = Rank b & M <==> W )

thus ( b is_cofinal_with omega & M = Rank b & M <==> W ) by A1, Th9; :: thesis: verum

( a is_cofinal_with omega & M = Rank a & M <==> W ) )

assume omega in W ; :: thesis: ex a being Ordinal of W ex M being non empty set st

( a is_cofinal_with omega & M = Rank a & M <==> W )

then consider b being Ordinal of W, M being non empty set such that

A1: ( b is_cofinal_with omega & M = Rank b & M is_elementary_subsystem_of W ) by Th35;

take b ; :: thesis: ex M being non empty set st

( b is_cofinal_with omega & M = Rank b & M <==> W )

take M ; :: thesis: ( b is_cofinal_with omega & M = Rank b & M <==> W )

thus ( b is_cofinal_with omega & M = Rank b & M <==> W ) by A1, Th9; :: thesis: verum