Theorem axhilex-zf 31141

Description: Derive Axiom ax-hilex 31159 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 31129	. 2 ⊢ ℋ = (BaseSet‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)
2	1	hlex 31058	1 ⊢ ℋ ∈ V

Syntax hints:   = wceq 1559   ∈ wcel 2141  Vcvv 3453  ⟨cop 4585  CHil_OLDchlo 31045   ℋchba 31079   +_ℎ cva 31080   ·_ℎ csm 31081  norm_ℎcno 31083

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-nul 5253

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-v 3455  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-sn 4580  df-pr 4582  df-uni 4863  df-iota 6472  df-fv 6524  df-hba 31129

This theorem is referenced by: (None)