![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
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 4423 | . . 3 ⊢ ∅ ⊆ (Atoms‘𝐾) | |
2 | 1 | a1i 11 | . 2 ⊢ (𝐾 ∈ HL → ∅ ⊆ (Atoms‘𝐾)) |
3 | eqid 2740 | . . 3 ⊢ (⊥𝑃‘𝐾) = (⊥𝑃‘𝐾) | |
4 | 3 | 2pol0N 39870 | . 2 ⊢ (𝐾 ∈ HL → ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘∅)) = ∅) |
5 | eqid 2740 | . . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
6 | 0psubcl.c | . . 3 ⊢ 𝐶 = (PSubCl‘𝐾) | |
7 | 5, 3, 6 | ispsubclN 39896 | . 2 ⊢ (𝐾 ∈ HL → (∅ ∈ 𝐶 ↔ (∅ ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘∅)) = ∅))) |
8 | 2, 4, 7 | mpbir2and 712 | 1 ⊢ (𝐾 ∈ HL → ∅ ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2108 ⊆ wss 3976 ∅c0 4352 ‘cfv 6575 Atomscatm 39221 HLchlt 39308 ⊥𝑃cpolN 39861 PSubClcpscN 39893 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7772 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-f1 6580 df-fo 6581 df-f1o 6582 df-fv 6583 df-riota 7406 df-ov 7453 df-oprab 7454 df-proset 18367 df-poset 18385 df-plt 18402 df-lub 18418 df-glb 18419 df-join 18420 df-meet 18421 df-p0 18497 df-p1 18498 df-lat 18504 df-clat 18571 df-oposet 39134 df-ol 39136 df-oml 39137 df-covers 39224 df-ats 39225 df-atl 39256 df-cvlat 39280 df-hlat 39309 df-pmap 39463 df-polarityN 39862 df-psubclN 39894 |
This theorem is referenced by: pclfinclN 39909 |
Copyright terms: Public domain | W3C validator |