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Theorem axhilex-zf 21553
Description: Derive axiom ax-hilex 21571 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
StepHypRef Expression
1 df-hba 21541 . 2  |-  ~H  =  ( BaseSet `  <. <.  +h  ,  .h  >. ,  normh >. )
21hlex 21469 1  |-  ~H  e.  _V
Colors of variables: wff set class
Syntax hints:    = wceq 1624    e. wcel 1685   _Vcvv 2789   <.cop 3644   CHil OLDchlo 21456   ~Hchil 21491    +h cva 21492    .h csm 21493   normhcno 21495
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-rex 2550  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-sn 3647  df-pr 3648  df-uni 3829  df-fv 5229  df-hba 21541
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