Theorem 0psubN 38925

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 4397	. . 3 ⊢ ∅ ⊆ (Atoms‘𝐾)
2		ral0 4513	. . 3 ⊢ ∀𝑝 ∈ ∅ ∀𝑞 ∈ ∅ ∀𝑟 ∈ (Atoms‘𝐾)(𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ ∅)
3	1, 2	pm3.2i 469	. 2 ⊢ (∅ ⊆ (Atoms‘𝐾) ∧ ∀𝑝 ∈ ∅ ∀𝑞 ∈ ∅ ∀𝑟 ∈ (Atoms‘𝐾)(𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ ∅))
4		eqid 2730	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
5		eqid 2730	. . 3 ⊢ (join‘𝐾) = (join‘𝐾)
6		eqid 2730	. . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
7		0psub.s	. . 3 ⊢ 𝑆 = (PSubSp‘𝐾)
8	4, 5, 6, 7	ispsubsp 38921	. 2 ⊢ (𝐾 ∈ 𝑉 → (∅ ∈ 𝑆 ↔ (∅ ⊆ (Atoms‘𝐾) ∧ ∀𝑝 ∈ ∅ ∀𝑞 ∈ ∅ ∀𝑟 ∈ (Atoms‘𝐾)(𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ ∅))))
9	3, 8	mpbiri 257	1 ⊢ (𝐾 ∈ 𝑉 → ∅ ∈ 𝑆)

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1539   ∈ wcel 2104  ∀wral 3059   ⊆ wss 3949  ∅c0 4323   class class class wbr 5149  ‘cfv 6544  (class class class)co 7413  lecple 17210  joincjn 18270  Atomscatm 38438  PSubSpcpsubsp 38672

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7416  df-psubsp 38679

This theorem is referenced by: (None)