Mathbox for Norm Megill < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  isat2 Structured version   Visualization version   GIF version

Theorem isat2 35096
 Description: The predicate "is an atom". (elatcv0 29540 analog.) (Contributed by NM, 18-Jun-2012.)
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
isat2 ((𝐾𝐷𝑃𝐵) → (𝑃𝐴0 𝐶𝑃))

Proof of Theorem isat2
StepHypRef Expression
1 isatom.b . . 3 𝐵 = (Base‘𝐾)
2 isatom.z . . 3 0 = (0.‘𝐾)
3 isatom.c . . 3 𝐶 = ( ⋖ ‘𝐾)
4 isatom.a . . 3 𝐴 = (Atoms‘𝐾)
51, 2, 3, 4isat 35095 . 2 (𝐾𝐷 → (𝑃𝐴 ↔ (𝑃𝐵0 𝐶𝑃)))
65baibd 529 1 ((𝐾𝐷𝑃𝐵) → (𝑃𝐴0 𝐶𝑃))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   = wceq 1631   ∈ wcel 2145   class class class wbr 4786  ‘cfv 6031  Basecbs 16064  0.cp0 17245   ⋖ ccvr 35071  Atomscatm 35072 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-ats 35076 This theorem is referenced by:  llncvrlpln  35366  lplncvrlvol  35424  lhpm0atN  35837
 Copyright terms: Public domain W3C validator