Theorem 0psubN 39751

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 4400	. . 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 470	. 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)
8	4, 5, 6, 7	ispsubsp 39747	. 2 ⊢ (𝐾 ∈ 𝑉 → (∅ ∈ 𝑆 ↔ (∅ ⊆ (Atoms‘𝐾) ∧ ∀𝑝 ∈ ∅ ∀𝑞 ∈ ∅ ∀𝑟 ∈ (Atoms‘𝐾)(𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ ∅))))
9	3, 8	mpbiri 258	1 ⊢ (𝐾 ∈ 𝑉 → ∅ ∈ 𝑆)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3061   ⊆ wss 3951  ∅c0 4333   class class class wbr 5143  ‘cfv 6561  (class class class)co 7431  lecple 17304  joincjn 18357  Atomscatm 39264  PSubSpcpsubsp 39498

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-psubsp 39505

This theorem is referenced by: (None)