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Theorem iswatN 39997
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 39996 . . 3 ((𝐾𝐵𝐷𝐴) → (𝑊𝐷) = (𝐴 ∖ ((⊥𝑃𝐾)‘{𝐷})))
54eleq2d 2826 . 2 ((𝐾𝐵𝐷𝐴) → (𝑃 ∈ (𝑊𝐷) ↔ 𝑃 ∈ (𝐴 ∖ ((⊥𝑃𝐾)‘{𝐷}))))
6 eldif 3960 . 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 1539  wcel 2107  cdif 3947  {csn 4625  cfv 6560  Atomscatm 39265  𝑃cpolN 39905  WAtomscwpointsN 39989
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-watsN 39993
This theorem is referenced by: (None)
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