Theorem axhilex-zf 22437

Description: Derive axiom ax-hilex 22455 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. CHil OLD

Assertion

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

Proof of Theorem axhilex-zf

Step	Hyp	Ref	Expression
1		df-hba 22425	. 2 |- ~H = ( BaseSet ` <. <. +h , .h >. , normh >. )
2	1	hlex 22353	1 |- ~H e. _V

Syntax hints:   = wceq 1649   e. wcel 1721   _Vcvv 2916   <.cop 3777   CHil OLDchlo 22340   ~Hchil 22375   +h cva 22376   .h csm 22377   normhcno 22379

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-nul 4298

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-sn 3780  df-pr 3781  df-uni 3976  df-iota 5377  df-fv 5421  df-hba 22425