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Theorem 0psubN 38668
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 = (PSubSpβ€˜πΎ)
Assertion
Ref Expression
0psubN (𝐾 ∈ 𝑉 β†’ βˆ… ∈ 𝑆)

Proof of Theorem 0psubN
Dummy variables π‘ž 𝑝 π‘Ÿ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0ss 4397 . . 3 βˆ… βŠ† (Atomsβ€˜πΎ)
2 ral0 4513 . . 3 βˆ€π‘ ∈ βˆ… βˆ€π‘ž ∈ βˆ… βˆ€π‘Ÿ ∈ (Atomsβ€˜πΎ)(π‘Ÿ(leβ€˜πΎ)(𝑝(joinβ€˜πΎ)π‘ž) β†’ π‘Ÿ ∈ βˆ…)
31, 2pm3.2i 472 . 2 (βˆ… βŠ† (Atomsβ€˜πΎ) ∧ βˆ€π‘ ∈ βˆ… βˆ€π‘ž ∈ βˆ… βˆ€π‘Ÿ ∈ (Atomsβ€˜πΎ)(π‘Ÿ(leβ€˜πΎ)(𝑝(joinβ€˜πΎ)π‘ž) β†’ π‘Ÿ ∈ βˆ…))
4 eqid 2733 . . 3 (leβ€˜πΎ) = (leβ€˜πΎ)
5 eqid 2733 . . 3 (joinβ€˜πΎ) = (joinβ€˜πΎ)
6 eqid 2733 . . 3 (Atomsβ€˜πΎ) = (Atomsβ€˜πΎ)
7 0psub.s . . 3 𝑆 = (PSubSpβ€˜πΎ)
84, 5, 6, 7ispsubsp 38664 . 2 (𝐾 ∈ 𝑉 β†’ (βˆ… ∈ 𝑆 ↔ (βˆ… βŠ† (Atomsβ€˜πΎ) ∧ βˆ€π‘ ∈ βˆ… βˆ€π‘ž ∈ βˆ… βˆ€π‘Ÿ ∈ (Atomsβ€˜πΎ)(π‘Ÿ(leβ€˜πΎ)(𝑝(joinβ€˜πΎ)π‘ž) β†’ π‘Ÿ ∈ βˆ…))))
93, 8mpbiri 258 1 (𝐾 ∈ 𝑉 β†’ βˆ… ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062   βŠ† wss 3949  βˆ…c0 4323   class class class wbr 5149  β€˜cfv 6544  (class class class)co 7409  lecple 17204  joincjn 18264  Atomscatm 38181  PSubSpcpsubsp 38415
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  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 7412  df-psubsp 38422
This theorem is referenced by: (None)
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