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Theorem iswatN 38016
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 38015 . . 3 ((𝐾𝐵𝐷𝐴) → (𝑊𝐷) = (𝐴 ∖ ((⊥𝑃𝐾)‘{𝐷})))
54eleq2d 2824 . 2 ((𝐾𝐵𝐷𝐴) → (𝑃 ∈ (𝑊𝐷) ↔ 𝑃 ∈ (𝐴 ∖ ((⊥𝑃𝐾)‘{𝐷}))))
6 eldif 3896 . 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 205  wa 396   = wceq 1539  wcel 2106  cdif 3883  {csn 4561  cfv 6426  Atomscatm 37285  𝑃cpolN 37924  WAtomscwpointsN 38008
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5208  ax-sep 5221  ax-nul 5228  ax-pr 5350
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3071  df-rab 3073  df-v 3431  df-sbc 3716  df-csb 3832  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5074  df-opab 5136  df-mpt 5157  df-id 5484  df-xp 5590  df-rel 5591  df-cnv 5592  df-co 5593  df-dm 5594  df-rn 5595  df-res 5596  df-ima 5597  df-iota 6384  df-fun 6428  df-fn 6429  df-f 6430  df-f1 6431  df-fo 6432  df-f1o 6433  df-fv 6434  df-watsN 38012
This theorem is referenced by: (None)
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