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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
StepHypRef Expression
1 df-hba 26634 . 2  |-  ~H  =  ( BaseSet `  <. <.  +h  ,  .h  >. ,  normh >. )
21hlex 26562 1  |-  ~H  e.  _V
Colors of variables: wff setvar class
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)
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