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Mirrors > Home > HSE Home > Th. List > axhilex-zf | Structured version Visualization version GIF version |
Description: Derive axiom ax-hilex 28770 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 28740 | . 2 ⊢ ℋ = (BaseSet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) | |
2 | 1 | hlex 28669 | 1 ⊢ ℋ ∈ V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2110 Vcvv 3495 〈cop 4567 CHilOLDchlo 28656 ℋchba 28690 +ℎ cva 28691 ·ℎ csm 28692 normℎcno 28694 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-nul 5203 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-sn 4562 df-pr 4564 df-uni 4833 df-iota 6309 df-fv 6358 df-hba 28740 |
This theorem is referenced by: (None) |
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