Theorem axhilex-zf 30951

Description: Derive Axiom ax-hilex 30969 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 ∈ CHilOLD

Assertion

Ref	Expression
axhilex-zf	⊢ ℋ ∈ V

Proof of Theorem axhilex-zf

Step	Hyp	Ref	Expression
1		df-hba 30939	. 2 ⊢ ℋ = (BaseSet‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)
2	1	hlex 30868	1 ⊢ ℋ ∈ V

Syntax hints:   = wceq 1541   ∈ wcel 2110  Vcvv 3434  ⟨cop 4580  CHil_OLDchlo 30855   ℋchba 30889   +_ℎ cva 30890   ·_ℎ csm 30891  norm_ℎcno 30893

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-ext 2702  ax-nul 5242

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-v 3436  df-dif 3903  df-un 3905  df-ss 3917  df-nul 4282  df-sn 4575  df-pr 4577  df-uni 4858  df-iota 6433  df-fv 6485  df-hba 30939

This theorem is referenced by: (None)