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Theorem isat 35094
Description: The predicate "is an atom". (ela 29528 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 35093 . . 3 (𝐾𝐷𝐴 = {𝑥𝐵0 𝐶𝑥})
65eleq2d 2825 . 2 (𝐾𝐷 → (𝑃𝐴𝑃 ∈ {𝑥𝐵0 𝐶𝑥}))
7 breq2 4808 . . 3 (𝑥 = 𝑃 → ( 0 𝐶𝑥0 𝐶𝑃))
87elrab 3504 . 2 (𝑃 ∈ {𝑥𝐵0 𝐶𝑥} ↔ (𝑃𝐵0 𝐶𝑃))
96, 8syl6bb 276 1 (𝐾𝐷 → (𝑃𝐴 ↔ (𝑃𝐵0 𝐶𝑃)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  {crab 3054   class class class wbr 4804  cfv 6049  Basecbs 16079  0.cp0 17258  ccvr 35070  Atomscatm 35071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fv 6057  df-ats 35075
This theorem is referenced by:  isat2  35095  atcvr0  35096  atbase  35097  isat3  35115  1cvrco  35279  1cvrjat  35282  ltrnatb  35944
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