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Theorem isat 39287
Description: The predicate "is an atom". (ela 32358 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 39286 . . 3 (𝐾𝐷𝐴 = {𝑥𝐵0 𝐶𝑥})
65eleq2d 2827 . 2 (𝐾𝐷 → (𝑃𝐴𝑃 ∈ {𝑥𝐵0 𝐶𝑥}))
7 breq2 5147 . . 3 (𝑥 = 𝑃 → ( 0 𝐶𝑥0 𝐶𝑃))
87elrab 3692 . 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 1540  wcel 2108  {crab 3436   class class class wbr 5143  cfv 6561  Basecbs 17247  0.cp0 18468  ccvr 39263  Atomscatm 39264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ats 39268
This theorem is referenced by:  isat2  39288  atcvr0  39289  atbase  39290  isat3  39308  1cvrco  39474  1cvrjat  39477  ltrnatb  40139
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