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Theorem 0psubN 36879
 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 4350 . . 3 ∅ ⊆ (Atoms‘𝐾)
2 ral0 4456 . . 3 𝑝 ∈ ∅ ∀𝑞 ∈ ∅ ∀𝑟 ∈ (Atoms‘𝐾)(𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ ∅)
31, 2pm3.2i 473 . 2 (∅ ⊆ (Atoms‘𝐾) ∧ ∀𝑝 ∈ ∅ ∀𝑞 ∈ ∅ ∀𝑟 ∈ (Atoms‘𝐾)(𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ ∅))
4 eqid 2821 . . 3 (le‘𝐾) = (le‘𝐾)
5 eqid 2821 . . 3 (join‘𝐾) = (join‘𝐾)
6 eqid 2821 . . 3 (Atoms‘𝐾) = (Atoms‘𝐾)
7 0psub.s . . 3 𝑆 = (PSubSp‘𝐾)
84, 5, 6, 7ispsubsp 36875 . 2 (𝐾𝑉 → (∅ ∈ 𝑆 ↔ (∅ ⊆ (Atoms‘𝐾) ∧ ∀𝑝 ∈ ∅ ∀𝑞 ∈ ∅ ∀𝑟 ∈ (Atoms‘𝐾)(𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ ∅))))
93, 8mpbiri 260 1 (𝐾𝑉 → ∅ ∈ 𝑆)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ∀wral 3138   ⊆ wss 3936  ∅c0 4291   class class class wbr 5059  ‘cfv 6350  (class class class)co 7150  lecple 16566  joincjn 17548  Atomscatm 36393  PSubSpcpsubsp 36626 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-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  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-rab 3147  df-v 3497  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-iota 6309  df-fun 6352  df-fv 6358  df-ov 7153  df-psubsp 36633 This theorem is referenced by: (None)
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