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Theorem axhilex-zf 31010
Description: Derive Axiom ax-hilex 31028 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 30998 . 2 ℋ = (BaseSet‘⟨⟨ + , · ⟩, norm⟩)
21hlex 30927 1 ℋ ∈ V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wcel 2106  Vcvv 3478  cop 4637  CHilOLDchlo 30914  chba 30948   + cva 30949   · csm 30950  normcno 30952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-nul 5312
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-sn 4632  df-pr 4634  df-uni 4913  df-iota 6516  df-fv 6571  df-hba 30998
This theorem is referenced by: (None)
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