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Mirrors > Home > HSE Home > Th. List > axhilex-zf | Structured version Visualization version GIF version |
Description: Derive Axiom ax-hilex 29827 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 29797 | . 2 ⊢ ℋ = (BaseSet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) | |
2 | 1 | hlex 29726 | 1 ⊢ ℋ ∈ V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∈ wcel 2106 Vcvv 3443 〈cop 4590 CHilOLDchlo 29713 ℋchba 29747 +ℎ cva 29748 ·ℎ csm 29749 normℎcno 29751 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 ax-nul 5261 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-ne 2942 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-sn 4585 df-pr 4587 df-uni 4864 df-iota 6445 df-fv 6501 df-hba 29797 |
This theorem is referenced by: (None) |
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