Theorem axhilex-zf 27838

Description: Derive axiom ax-hilex 27856 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Hypotheses

Ref	Expression
axhil.1	|- U = <. <. +h , .h >. , normh >.
axhil.2	|- U e. CHilOLD

Assertion

Ref	Expression
axhilex-zf	|- ~H e. _V

Proof of Theorem axhilex-zf

Step	Hyp	Ref	Expression
1		df-hba 27826	. 2 |- ~H = ( BaseSet ` <. <. +h , .h >. , normh >. )
2	1	hlex 27754	1 |- ~H e. _V

Syntax hints:   = wceq 1483   e. wcel 1990   _Vcvv 3200   <.cop 4183   CHilOLDchlo 27741   ~Hchil 27776   +h cva 27777   .h csm 27778   normhcno 27780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-sn 4178  df-pr 4180  df-uni 4437  df-iota 5851  df-fv 5896  df-hba 27826

This theorem is referenced by: (None)