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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
StepHypRef Expression
1 df-hba 26100 . 2  |-  ~H  =  ( BaseSet `  <. <.  +h  ,  .h  >. ,  normh >. )
21hlex 26028 1  |-  ~H  e.  _V
Colors of variables: wff setvar class
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)
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