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Theorem axhilex-zf 21391
Description: Derive axiom ax-hilex 21409 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 21379 . 2  |-  ~H  =  ( BaseSet `  <. <.  +h  ,  .h  >. ,  normh >. )
21hlex 21307 1  |-  ~H  e.  _V
Colors of variables: wff set class
Syntax hints:    = wceq 1619    e. wcel 1621   _Vcvv 2727   <.cop 3547   CHil OLDchlo 21294   ~Hchil 21329    +h cva 21330    .h csm 21331   normhcno 21333
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-rex 2514  df-v 2729  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-sn 3550  df-pr 3551  df-uni 3728  df-fv 4608  df-hba 21379
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