Theorem axhilex-zf 30972

Description: Derive Axiom ax-hilex 30990 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 30960	. 2 ⊢ ℋ = (BaseSet‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)
2	1	hlex 30889	1 ⊢ ℋ ∈ V

Syntax hints:   = wceq 1541   ∈ wcel 2113  Vcvv 3438  ⟨cop 4583  CHil_OLDchlo 30876   ℋchba 30910   +_ℎ cva 30911   ·_ℎ csm 30912  norm_ℎcno 30914

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 2115  ax-9 2123  ax-ext 2705  ax-nul 5248

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 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2931  df-v 3440  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4285  df-sn 4578  df-pr 4580  df-uni 4861  df-iota 6445  df-fv 6497  df-hba 30960

This theorem is referenced by: (None)