Theorem axhilex-zf 31333

Description: Derive Axiom ax-hilex 31351 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 31321	. 2 ⊢ ℋ = (BaseSet‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)
2	1	hlex 31250	1 ⊢ ℋ ∈ V

Syntax hints:   = wceq 1570   ∈ wcel 2143  Vcvv 3455  ⟨cop 4595  CHil_OLDchlo 31237   ℋchba 31271   +_ℎ cva 31272   ·_ℎ csm 31273  norm_ℎcno 31275

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-nul 5269

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-v 3457  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-sn 4590  df-pr 4592  df-uni 4873  df-iota 6492  df-fv 6544  df-hba 31321

This theorem is referenced by: (None)