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Theorem psubcliN 39451
Description: Property of a closed projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
psubclset.a 𝐴 = (Atomsβ€˜πΎ)
psubclset.p βŠ₯ = (βŠ₯π‘ƒβ€˜πΎ)
psubclset.c 𝐢 = (PSubClβ€˜πΎ)
Assertion
Ref Expression
psubcliN ((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐢) β†’ (𝑋 βŠ† 𝐴 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋))

Proof of Theorem psubcliN
StepHypRef Expression
1 psubclset.a . . 3 𝐴 = (Atomsβ€˜πΎ)
2 psubclset.p . . 3 βŠ₯ = (βŠ₯π‘ƒβ€˜πΎ)
3 psubclset.c . . 3 𝐢 = (PSubClβ€˜πΎ)
41, 2, 3ispsubclN 39450 . 2 (𝐾 ∈ 𝐷 β†’ (𝑋 ∈ 𝐢 ↔ (𝑋 βŠ† 𝐴 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋)))
54biimpa 475 1 ((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐢) β†’ (𝑋 βŠ† 𝐴 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098   βŠ† wss 3949  β€˜cfv 6553  Atomscatm 38775  βŠ₯𝑃cpolN 39415  PSubClcpscN 39447
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6505  df-fun 6555  df-fv 6561  df-psubclN 39448
This theorem is referenced by:  psubclsubN  39453  psubclssatN  39454
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