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Theorem 0psubN 37505
Description: The empty set is a projective subspace. Remark below Definition 15.1 of [MaedaMaeda] p. 61. (Contributed by NM, 13-Oct-2011.) (New usage is discouraged.)
Hypothesis
Ref Expression
0psub.s 𝑆 = (PSubSp‘𝐾)
Assertion
Ref Expression
0psubN (𝐾𝑉 → ∅ ∈ 𝑆)

Proof of Theorem 0psubN
Dummy variables 𝑞 𝑝 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0ss 4316 . . 3 ∅ ⊆ (Atoms‘𝐾)
2 ral0 4429 . . 3 𝑝 ∈ ∅ ∀𝑞 ∈ ∅ ∀𝑟 ∈ (Atoms‘𝐾)(𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ ∅)
31, 2pm3.2i 474 . 2 (∅ ⊆ (Atoms‘𝐾) ∧ ∀𝑝 ∈ ∅ ∀𝑞 ∈ ∅ ∀𝑟 ∈ (Atoms‘𝐾)(𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ ∅))
4 eqid 2737 . . 3 (le‘𝐾) = (le‘𝐾)
5 eqid 2737 . . 3 (join‘𝐾) = (join‘𝐾)
6 eqid 2737 . . 3 (Atoms‘𝐾) = (Atoms‘𝐾)
7 0psub.s . . 3 𝑆 = (PSubSp‘𝐾)
84, 5, 6, 7ispsubsp 37501 . 2 (𝐾𝑉 → (∅ ∈ 𝑆 ↔ (∅ ⊆ (Atoms‘𝐾) ∧ ∀𝑝 ∈ ∅ ∀𝑞 ∈ ∅ ∀𝑟 ∈ (Atoms‘𝐾)(𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ ∅))))
93, 8mpbiri 261 1 (𝐾𝑉 → ∅ ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1543  wcel 2110  wral 3061  wss 3871  c0 4242   class class class wbr 5058  cfv 6385  (class class class)co 7218  lecple 16814  joincjn 17823  Atomscatm 37019  PSubSpcpsubsp 37252
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 2708  ax-sep 5197  ax-nul 5204  ax-pow 5263  ax-pr 5327
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3415  df-dif 3874  df-un 3876  df-in 3878  df-ss 3888  df-nul 4243  df-if 4445  df-pw 4520  df-sn 4547  df-pr 4549  df-op 4553  df-uni 4825  df-br 5059  df-opab 5121  df-mpt 5141  df-id 5460  df-xp 5562  df-rel 5563  df-cnv 5564  df-co 5565  df-dm 5566  df-iota 6343  df-fun 6387  df-fv 6393  df-ov 7221  df-psubsp 37259
This theorem is referenced by: (None)
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