Theorem axhilex-zf 26112

Description: Derive axiom ax-hilex 26130 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 26100	. 2 |- ~H = ( BaseSet ` <. <. +h , .h >. , normh >. )
2	1	hlex 26028	1 |- ~H e. _V

Syntax hints:   = wceq 1395   e. wcel 1819   _Vcvv 3109   <.cop 4038   CHilOLDchlo 26015   ~Hchil 26050   +h cva 26051   .h csm 26052   normhcno 26054

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-nul 4586

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-sn 4033  df-pr 4035  df-uni 4252  df-iota 5557  df-fv 5602  df-hba 26100

This theorem is referenced by: (None)