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Mirrors > Home > HSE Home > Th. List > axhilex-zf | Structured version Visualization version GIF version |
Description: Derive Axiom ax-hilex 29052 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 29022 | . 2 ⊢ ℋ = (BaseSet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) | |
2 | 1 | hlex 28951 | 1 ⊢ ℋ ∈ V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∈ wcel 2110 Vcvv 3401 〈cop 4537 CHilOLDchlo 28938 ℋchba 28972 +ℎ cva 28973 ·ℎ csm 28974 normℎcno 28976 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-nul 5188 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-ral 3059 df-rex 3060 df-v 3403 df-sbc 3688 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-sn 4532 df-pr 4534 df-uni 4810 df-iota 6327 df-fv 6377 df-hba 29022 |
This theorem is referenced by: (None) |
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