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| Description: Derive Axiom ax-hilex 31019 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| axhil.1 | ⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 | 
| axhil.2 | ⊢ 𝑈 ∈ CHilOLD | 
| Ref | Expression | 
|---|---|
| axhilex-zf | ⊢ ℋ ∈ V | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-hba 30989 | . 2 ⊢ ℋ = (BaseSet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) | |
| 2 | 1 | hlex 30918 | 1 ⊢ ℋ ∈ V | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1539 ∈ wcel 2107 Vcvv 3479 〈cop 4631 CHilOLDchlo 30905 ℋchba 30939 +ℎ cva 30940 ·ℎ csm 30941 normℎcno 30943 | 
| 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 2707 ax-nul 5305 | 
| 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 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-sn 4626 df-pr 4628 df-uni 4907 df-iota 6513 df-fv 6568 df-hba 30989 | 
| This theorem is referenced by: (None) | 
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