Theorem axhilex-zf 26646

Description: Derive axiom ax-hilex 26664 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 26634	. 2 |- ~H = ( BaseSet ` <. <. +h , .h >. , normh >. )
2	1	hlex 26562	1 |- ~H e. _V

Syntax hints:   = wceq 1446   e. wcel 1889   _Vcvv 3047   <.cop 3976   CHilOLDchlo 26549   ~Hchil 26584   +h cva 26585   .h csm 26586   normhcno 26588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-nul 4537

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-v 3049  df-sbc 3270  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-sn 3971  df-pr 3973  df-uni 4202  df-iota 5549  df-fv 5593  df-hba 26634

This theorem is referenced by: (None)