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Theorem 0psubN 39146
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 4392 . . 3 ∅ ⊆ (Atoms‘𝐾)
2 ral0 4508 . . 3 𝑝 ∈ ∅ ∀𝑞 ∈ ∅ ∀𝑟 ∈ (Atoms‘𝐾)(𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ ∅)
31, 2pm3.2i 470 . 2 (∅ ⊆ (Atoms‘𝐾) ∧ ∀𝑝 ∈ ∅ ∀𝑞 ∈ ∅ ∀𝑟 ∈ (Atoms‘𝐾)(𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ ∅))
4 eqid 2727 . . 3 (le‘𝐾) = (le‘𝐾)
5 eqid 2727 . . 3 (join‘𝐾) = (join‘𝐾)
6 eqid 2727 . . 3 (Atoms‘𝐾) = (Atoms‘𝐾)
7 0psub.s . . 3 𝑆 = (PSubSp‘𝐾)
84, 5, 6, 7ispsubsp 39142 . 2 (𝐾𝑉 → (∅ ∈ 𝑆 ↔ (∅ ⊆ (Atoms‘𝐾) ∧ ∀𝑝 ∈ ∅ ∀𝑞 ∈ ∅ ∀𝑟 ∈ (Atoms‘𝐾)(𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ ∅))))
93, 8mpbiri 258 1 (𝐾𝑉 → ∅ ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534  wcel 2099  wral 3056  wss 3944  c0 4318   class class class wbr 5142  cfv 6542  (class class class)co 7414  lecple 17225  joincjn 18288  Atomscatm 38659  PSubSpcpsubsp 38893
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6494  df-fun 6544  df-fv 6550  df-ov 7417  df-psubsp 38900
This theorem is referenced by: (None)
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