Theorem psubcliN 40235

Description: Property of a closed projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)

Hypotheses

Ref	Expression
psubclset.a	⊢ 𝐴 = (Atoms‘𝐾)
psubclset.p	⊢ ⊥ = (⊥𝑃‘𝐾)
psubclset.c	⊢ 𝐶 = (PSubCl‘𝐾)

Assertion

Ref	Expression
psubcliN	⊢ ((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐶) → (𝑋 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋))

Proof of Theorem psubcliN

Step	Hyp	Ref	Expression
1		psubclset.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
2		psubclset.p	. . 3 ⊢ ⊥ = (⊥𝑃‘𝐾)
3		psubclset.c	. . 3 ⊢ 𝐶 = (PSubCl‘𝐾)
4	1, 2, 3	ispsubclN 40234	. 2 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝐶 ↔ (𝑋 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)))
5	4	biimpa 476	1 ⊢ ((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐶) → (𝑋 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ⊆ wss 3902  ‘cfv 6493  Atomscatm 39560  ⊥_𝑃cpolN 40199  PSubClcpscN 40231

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-iota 6449  df-fun 6495  df-fv 6501  df-psubclN 40232

This theorem is referenced by:  psubclsubN  40237  psubclssatN  40238