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Theorem isat 39268
Description: The predicate "is an atom". (ela 32368 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 39267 . . 3 (𝐾𝐷𝐴 = {𝑥𝐵0 𝐶𝑥})
65eleq2d 2825 . 2 (𝐾𝐷 → (𝑃𝐴𝑃 ∈ {𝑥𝐵0 𝐶𝑥}))
7 breq2 5152 . . 3 (𝑥 = 𝑃 → ( 0 𝐶𝑥0 𝐶𝑃))
87elrab 3695 . 2 (𝑃 ∈ {𝑥𝐵0 𝐶𝑥} ↔ (𝑃𝐵0 𝐶𝑃))
96, 8bitrdi 287 1 (𝐾𝐷 → (𝑃𝐴 ↔ (𝑃𝐵0 𝐶𝑃)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  {crab 3433   class class class wbr 5148  cfv 6563  Basecbs 17245  0.cp0 18481  ccvr 39244  Atomscatm 39245
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-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-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  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-iota 6516  df-fun 6565  df-fv 6571  df-ats 39249
This theorem is referenced by:  isat2  39269  atcvr0  39270  atbase  39271  isat3  39289  1cvrco  39455  1cvrjat  39458  ltrnatb  40120
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