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Theorem iswatN 39977
Description: The predicate "is a W atom" (corresponding to fiducial atom 𝐷). (Contributed by NM, 26-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
watomfval.a 𝐴 = (Atoms‘𝐾)
watomfval.p 𝑃 = (⊥𝑃𝐾)
watomfval.w 𝑊 = (WAtoms‘𝐾)
Assertion
Ref Expression
iswatN ((𝐾𝐵𝐷𝐴) → (𝑃 ∈ (𝑊𝐷) ↔ (𝑃𝐴 ∧ ¬ 𝑃 ∈ ((⊥𝑃𝐾)‘{𝐷}))))

Proof of Theorem iswatN
StepHypRef Expression
1 watomfval.a . . . 4 𝐴 = (Atoms‘𝐾)
2 watomfval.p . . . 4 𝑃 = (⊥𝑃𝐾)
3 watomfval.w . . . 4 𝑊 = (WAtoms‘𝐾)
41, 2, 3watvalN 39976 . . 3 ((𝐾𝐵𝐷𝐴) → (𝑊𝐷) = (𝐴 ∖ ((⊥𝑃𝐾)‘{𝐷})))
54eleq2d 2825 . 2 ((𝐾𝐵𝐷𝐴) → (𝑃 ∈ (𝑊𝐷) ↔ 𝑃 ∈ (𝐴 ∖ ((⊥𝑃𝐾)‘{𝐷}))))
6 eldif 3973 . 2 (𝑃 ∈ (𝐴 ∖ ((⊥𝑃𝐾)‘{𝐷})) ↔ (𝑃𝐴 ∧ ¬ 𝑃 ∈ ((⊥𝑃𝐾)‘{𝐷})))
75, 6bitrdi 287 1 ((𝐾𝐵𝐷𝐴) → (𝑃 ∈ (𝑊𝐷) ↔ (𝑃𝐴 ∧ ¬ 𝑃 ∈ ((⊥𝑃𝐾)‘{𝐷}))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  cdif 3960  {csn 4631  cfv 6563  Atomscatm 39245  𝑃cpolN 39885  WAtomscwpointsN 39969
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-watsN 39973
This theorem is referenced by: (None)
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