Theorem axhfi-zf

Description: Derive Axiom ax-hfi from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008) (New usage is discouraged.)

	Ref	Expression
Hypotheses	axhil.1	|- U = <. <. +h , .h >. , normh >.
	axhil.2	|- U e. CHilOLD
	axhfi.1	|- .ih = ( .iOLD ` U )
Assertion	axhfi-zf	|- .ih : ( ~H X. ~H ) --> CC

Proof

Step	Hyp	Ref	Expression
1		axhil.1	|- U = <. <. +h , .h >. , normh >.
2		axhil.2	|- U e. CHilOLD
3		axhfi.1	|- .ih = ( .iOLD ` U )
4		df-hba	|- ~H = ( BaseSet ` <. <. +h , .h >. , normh >. )
5	1	fveq2i	|- ( BaseSet ` U ) = ( BaseSet ` <. <. +h , .h >. , normh >. )
6	4 5	eqtr4i	|- ~H = ( BaseSet ` U )
7	6 3	hlipf	|- ( U e. CHilOLD -> .ih : ( ~H X. ~H ) --> CC )
8	2 7	ax-mp	|- .ih : ( ~H X. ~H ) --> CC