Theorem axhilex-zf 31462

Description: Derive Axiom ax-hilex 31480 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 31450	. 2 ⊢ ℋ = (BaseSet‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)
2	1	hlex 31379	1 ⊢ ℋ ∈ V

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2145  Vcvv 3450  ⟨cop 4590  CHil_OLDchlo 31366   ℋchba 31400   +_ℎ cva 31401   ·_ℎ csm 31402  norm_ℎcno 31404

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-nul 5263

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-v 3452  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-sn 4585  df-pr 4587  df-uni 4868  df-iota 6489  df-fv 6541  df-hba 31450

This theorem is used by: (None)