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Theorem isat 36421
Description: The predicate "is an atom". (ela 30115 analog.) (Contributed by NM, 18-Sep-2011.)
Hypotheses
Ref Expression
isatom.b 𝐵 = (Base‘𝐾)
isatom.z 0 = (0.‘𝐾)
isatom.c 𝐶 = ( ⋖ ‘𝐾)
isatom.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
isat (𝐾𝐷 → (𝑃𝐴 ↔ (𝑃𝐵0 𝐶𝑃)))

Proof of Theorem isat
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 isatom.b . . . 4 𝐵 = (Base‘𝐾)
2 isatom.z . . . 4 0 = (0.‘𝐾)
3 isatom.c . . . 4 𝐶 = ( ⋖ ‘𝐾)
4 isatom.a . . . 4 𝐴 = (Atoms‘𝐾)
51, 2, 3, 4pats 36420 . . 3 (𝐾𝐷𝐴 = {𝑥𝐵0 𝐶𝑥})
65eleq2d 2898 . 2 (𝐾𝐷 → (𝑃𝐴𝑃 ∈ {𝑥𝐵0 𝐶𝑥}))
7 breq2 5069 . . 3 (𝑥 = 𝑃 → ( 0 𝐶𝑥0 𝐶𝑃))
87elrab 3679 . 2 (𝑃 ∈ {𝑥𝐵0 𝐶𝑥} ↔ (𝑃𝐵0 𝐶𝑃))
96, 8syl6bb 289 1 (𝐾𝐷 → (𝑃𝐴 ↔ (𝑃𝐵0 𝐶𝑃)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1533  wcel 2110  {crab 3142   class class class wbr 5065  cfv 6354  Basecbs 16482  0.cp0 17646  ccvr 36397  Atomscatm 36398
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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pr 5329
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-iota 6313  df-fun 6356  df-fv 6362  df-ats 36402
This theorem is referenced by:  isat2  36422  atcvr0  36423  atbase  36424  isat3  36442  1cvrco  36607  1cvrjat  36610  ltrnatb  37272
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