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

Theorem atssbase 34054
Description: The set of atoms is a subset of the base set. (atssch 29048 analog.) (Contributed by NM, 21-Oct-2011.)
Hypotheses
Ref Expression
atombase.b 𝐵 = (Base‘𝐾)
atombase.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
atssbase 𝐴𝐵

Proof of Theorem atssbase
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 atombase.b . . 3 𝐵 = (Base‘𝐾)
2 atombase.a . . 3 𝐴 = (Atoms‘𝐾)
31, 2atbase 34053 . 2 (𝑥𝐴𝑥𝐵)
43ssriv 3587 1 𝐴𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  wss 3555  cfv 5847  Basecbs 15781  Atomscatm 34027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fv 5855  df-ats 34031
This theorem is referenced by:  atlatmstc  34083  atlatle  34084  pmapssbaN  34523  pmaple  34524  polsubN  34670  2polvalN  34677  2polssN  34678  3polN  34679  2pmaplubN  34689  paddunN  34690  poldmj1N  34691  pnonsingN  34696  ispsubcl2N  34710  psubclinN  34711  paddatclN  34712  polsubclN  34715  poml4N  34716
  Copyright terms: Public domain W3C validator