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Theorem isat 35362
Description: The predicate "is an atom". (ela 29754 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 35361 . . 3 (𝐾𝐷𝐴 = {𝑥𝐵0 𝐶𝑥})
65eleq2d 2893 . 2 (𝐾𝐷 → (𝑃𝐴𝑃 ∈ {𝑥𝐵0 𝐶𝑥}))
7 breq2 4878 . . 3 (𝑥 = 𝑃 → ( 0 𝐶𝑥0 𝐶𝑃))
87elrab 3586 . 2 (𝑃 ∈ {𝑥𝐵0 𝐶𝑥} ↔ (𝑃𝐵0 𝐶𝑃))
96, 8syl6bb 279 1 (𝐾𝐷 → (𝑃𝐴 ↔ (𝑃𝐵0 𝐶𝑃)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1658  wcel 2166  {crab 3122   class class class wbr 4874  cfv 6124  Basecbs 16223  0.cp0 17391  ccvr 35338  Atomscatm 35339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pr 5128
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-sbc 3664  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-br 4875  df-opab 4937  df-mpt 4954  df-id 5251  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-iota 6087  df-fun 6126  df-fv 6132  df-ats 35343
This theorem is referenced by:  isat2  35363  atcvr0  35364  atbase  35365  isat3  35383  1cvrco  35548  1cvrjat  35551  ltrnatb  36213
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