Theorem axhilex-zf 30894

Description: Derive Axiom ax-hilex 30912 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 30882	. 2 ⊢ ℋ = (BaseSet‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)
2	1	hlex 30811	1 ⊢ ℋ ∈ V

Syntax hints:   = wceq 1539   ∈ wcel 2107  Vcvv 3457  ⟨cop 4605  CHil_OLDchlo 30798   ℋchba 30832   +_ℎ cva 30833   ·_ℎ csm 30834  norm_ℎcno 30836

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706  ax-nul 5273

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ne 2932  df-v 3459  df-dif 3927  df-un 3929  df-ss 3941  df-nul 4307  df-sn 4600  df-pr 4602  df-uni 4881  df-iota 6480  df-fv 6535  df-hba 30882

This theorem is referenced by: (None)