Theorem 0psubN 40378

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 4454	. . 3 ⊢ ∀𝑝 ∈ ∅ ∀𝑞 ∈ ∅ ∀𝑟 ∈ (Atoms‘𝐾)(𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ ∅)
3	1, 2	pm3.2i 474	. 2 ⊢ (∅ ⊆ (Atoms‘𝐾) ∧ ∀𝑝 ∈ ∅ ∀𝑞 ∈ ∅ ∀𝑟 ∈ (Atoms‘𝐾)(𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ ∅))
4		eqid 2764	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
5		eqid 2764	. . 3 ⊢ (join‘𝐾) = (join‘𝐾)
6		eqid 2764	. . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
7		0psub.s	. . 3 ⊢ 𝑆 = (PSubSp‘𝐾)
8	4, 5, 6, 7	ispsubsp 40374	. 2 ⊢ (𝐾 ∈ 𝑉 → (∅ ∈ 𝑆 ↔ (∅ ⊆ (Atoms‘𝐾) ∧ ∀𝑝 ∈ ∅ ∀𝑞 ∈ ∅ ∀𝑟 ∈ (Atoms‘𝐾)(𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ ∅))))
9	3, 8	mpbiri 260	1 ⊢ (𝐾 ∈ 𝑉 → ∅ ∈ 𝑆)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1562   ∈ wcel 2144  ∀wral 3078   ⊆ wss 3906  ∅c0 4287   class class class wbr 5102  ‘cfv 6523  (class class class)co 7398  lecple 17295  joincjn 18345  Atomscatm 39892  PSubSpcpsubsp 40125

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-iota 6479  df-fun 6525  df-fv 6531  df-ov 7401  df-psubsp 40132

This theorem is referenced by: (None)