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 4350 | . . 3 ⊢ ∅ ⊆ (Atoms‘𝐾) | |
2 | 1 | a1i 11 | . 2 ⊢ (𝐾 ∈ HL → ∅ ⊆ (Atoms‘𝐾)) |
3 | eqid 2821 | . . 3 ⊢ (⊥𝑃‘𝐾) = (⊥𝑃‘𝐾) | |
4 | 3 | 2pol0N 37062 | . 2 ⊢ (𝐾 ∈ HL → ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘∅)) = ∅) |
5 | eqid 2821 | . . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
6 | 0psubcl.c | . . 3 ⊢ 𝐶 = (PSubCl‘𝐾) | |
7 | 5, 3, 6 | ispsubclN 37088 | . 2 ⊢ (𝐾 ∈ HL → (∅ ∈ 𝐶 ↔ (∅ ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘∅)) = ∅))) |
8 | 2, 4, 7 | mpbir2and 711 | 1 ⊢ (𝐾 ∈ HL → ∅ ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ⊆ wss 3936 ∅c0 4291 ‘cfv 6355 Atomscatm 36414 HLchlt 36501 ⊥𝑃cpolN 37053 PSubClcpscN 37085 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-riotaBAD 36104 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-undef 7939 df-proset 17538 df-poset 17556 df-plt 17568 df-lub 17584 df-glb 17585 df-join 17586 df-meet 17587 df-p0 17649 df-p1 17650 df-lat 17656 df-clat 17718 df-oposet 36327 df-ol 36329 df-oml 36330 df-covers 36417 df-ats 36418 df-atl 36449 df-cvlat 36473 df-hlat 36502 df-pmap 36655 df-polarityN 37054 df-psubclN 37086 |
This theorem is referenced by: pclfinclN 37101 |
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