Theorem 0psubN 40551

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	⊢ (𝐾 ∈ 𝑉 → ∅ ∈ 𝑆)

Proof of Theorem 0psubN

Dummy variables 𝑞 𝑝 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0ss 4356	. . 3 ⊢ ∅ ⊆ (Atoms‘𝐾)
2		ral0 4458	. . 3 ⊢ ∀𝑝 ∈ ∅ ∀𝑞 ∈ ∅ ∀𝑟 ∈ (Atoms‘𝐾)(𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ ∅)
3	1, 2	pm3.2i 475	. 2 ⊢ (∅ ⊆ (Atoms‘𝐾) ∧ ∀𝑝 ∈ ∅ ∀𝑞 ∈ ∅ ∀𝑟 ∈ (Atoms‘𝐾)(𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ ∅))
4		eqid 2762	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
5		eqid 2762	. . 3 ⊢ (join‘𝐾) = (join‘𝐾)
6		eqid 2762	. . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
7		0psub.s	. . 3 ⊢ 𝑆 = (PSubSp‘𝐾)
8	4, 5, 6, 7	ispsubsp 40547	. 2 ⊢ (𝐾 ∈ 𝑉 → (∅ ∈ 𝑆 ↔ (∅ ⊆ (Atoms‘𝐾) ∧ ∀𝑝 ∈ ∅ ∀𝑞 ∈ ∅ ∀𝑟 ∈ (Atoms‘𝐾)(𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ ∅))))
9	3, 8	mpbiri 261	1 ⊢ (𝐾 ∈ 𝑉 → ∅ ∈ 𝑆)

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   = wceq 1569   ∈ wcel 2142  ∀wral 3078   ⊆ wss 3904  ∅c0 4285   class class class wbr 5108  ‘cfv 6536  (class class class)co 7412  lecple 17323  joincjn 18373  Atomscatm 40065  PSubSpcpsubsp 40298

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-iota 6492  df-fun 6538  df-fv 6544  df-ov 7415  df-psubsp 40305

This theorem is used by: (None)