Theorem axhilex-zf 29034

Description: Derive Axiom ax-hilex 29052 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 29022	. 2 ⊢ ℋ = (BaseSet‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)
2	1	hlex 28951	1 ⊢ ℋ ∈ V

Syntax hints:   = wceq 1543   ∈ wcel 2110  Vcvv 3401  ⟨cop 4537  CHil_OLDchlo 28938   ℋchba 28972   +_ℎ cva 28973   ·_ℎ csm 28974  norm_ℎcno 28976

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2706  ax-nul 5188

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2537  df-eu 2566  df-clab 2713  df-cleq 2726  df-clel 2812  df-ral 3059  df-rex 3060  df-v 3403  df-sbc 3688  df-dif 3860  df-un 3862  df-in 3864  df-ss 3874  df-nul 4228  df-sn 4532  df-pr 4534  df-uni 4810  df-iota 6327  df-fv 6377  df-hba 29022

This theorem is referenced by: (None)