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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 0psubclN | Structured version Visualization version GIF version |
Description: The empty set is a closed projective subspace. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
0psubcl.c | ⊢ 𝐶 = (PSubCl‘𝐾) |
Ref | Expression |
---|---|
0psubclN | ⊢ (𝐾 ∈ HL → ∅ ∈ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ss 4005 | . . 3 ⊢ ∅ ⊆ (Atoms‘𝐾) | |
2 | 1 | a1i 11 | . 2 ⊢ (𝐾 ∈ HL → ∅ ⊆ (Atoms‘𝐾)) |
3 | eqid 2651 | . . 3 ⊢ (⊥𝑃‘𝐾) = (⊥𝑃‘𝐾) | |
4 | 3 | 2pol0N 35515 | . 2 ⊢ (𝐾 ∈ HL → ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘∅)) = ∅) |
5 | eqid 2651 | . . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
6 | 0psubcl.c | . . 3 ⊢ 𝐶 = (PSubCl‘𝐾) | |
7 | 5, 3, 6 | ispsubclN 35541 | . 2 ⊢ (𝐾 ∈ HL → (∅ ∈ 𝐶 ↔ (∅ ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘∅)) = ∅))) |
8 | 2, 4, 7 | mpbir2and 977 | 1 ⊢ (𝐾 ∈ HL → ∅ ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1523 ∈ wcel 2030 ⊆ wss 3607 ∅c0 3948 ‘cfv 5926 Atomscatm 34868 HLchlt 34955 ⊥𝑃cpolN 35506 PSubClcpscN 35538 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-riotaBAD 34557 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-iun 4554 df-iin 4555 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-undef 7444 df-preset 16975 df-poset 16993 df-plt 17005 df-lub 17021 df-glb 17022 df-join 17023 df-meet 17024 df-p0 17086 df-p1 17087 df-lat 17093 df-clat 17155 df-oposet 34781 df-ol 34783 df-oml 34784 df-covers 34871 df-ats 34872 df-atl 34903 df-cvlat 34927 df-hlat 34956 df-pmap 35108 df-polarityN 35507 df-psubclN 35539 |
This theorem is referenced by: pclfinclN 35554 |
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