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Theorem axhilex-zf 29972
Description: Derive Axiom ax-hilex 29990 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
axhil.1 π‘ˆ = ⟨⟨ +β„Ž , Β·β„Ž ⟩, normβ„ŽβŸ©
axhil.2 π‘ˆ ∈ CHilOLD
Assertion
Ref Expression
axhilex-zf β„‹ ∈ V

Proof of Theorem axhilex-zf
StepHypRef Expression
1 df-hba 29960 . 2 β„‹ = (BaseSetβ€˜βŸ¨βŸ¨ +β„Ž , Β·β„Ž ⟩, normβ„ŽβŸ©)
21hlex 29889 1 β„‹ ∈ V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542   ∈ wcel 2107  Vcvv 3447  βŸ¨cop 4596  CHilOLDchlo 29876   β„‹chba 29910   +β„Ž cva 29911   Β·β„Ž csm 29912  normβ„Žcno 29914
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-nul 5267
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-sn 4591  df-pr 4593  df-uni 4870  df-iota 6452  df-fv 6508  df-hba 29960
This theorem is referenced by: (None)
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