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Theorem axhilex-zf 22485
Description: Derive axiom ax-hilex 22503 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 22473 . 2  |-  ~H  =  ( BaseSet `  <. <.  +h  ,  .h  >. ,  normh >. )
21hlex 22401 1  |-  ~H  e.  _V
Colors of variables: wff set class
Syntax hints:    = wceq 1653    e. wcel 1726   _Vcvv 2957   <.cop 3818   CHil OLDchlo 22388   ~Hchil 22423    +h cva 22424    .h csm 22425   normhcno 22427
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-nul 4339
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-sn 3821  df-pr 3822  df-uni 4017  df-iota 5419  df-fv 5463  df-hba 22473
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