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Theorem axhilex-zf 29809
Description: Derive Axiom ax-hilex 29827 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
axhil.1 𝑈 = ⟨⟨ + , · ⟩, norm
axhil.2 𝑈 ∈ CHilOLD
Assertion
Ref Expression
axhilex-zf ℋ ∈ V

Proof of Theorem axhilex-zf
StepHypRef Expression
1 df-hba 29797 . 2 ℋ = (BaseSet‘⟨⟨ + , · ⟩, norm⟩)
21hlex 29726 1 ℋ ∈ V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541  wcel 2106  Vcvv 3443  cop 4590  CHilOLDchlo 29713  chba 29747   + cva 29748   · csm 29749  normcno 29751
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-nul 5261
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2942  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-sn 4585  df-pr 4587  df-uni 4864  df-iota 6445  df-fv 6501  df-hba 29797
This theorem is referenced by: (None)
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