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 35064
Description: The set of atoms is a subset of the base set. (atssch 29524 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 35063 . 2 (𝑥𝐴𝑥𝐵)
43ssriv 3796 1 𝐴𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1637  wss 3763  cfv 6095  Basecbs 16062  Atomscatm 35037
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-8 2157  ax-9 2164  ax-10 2184  ax-11 2200  ax-12 2213  ax-13 2419  ax-ext 2781  ax-sep 4968  ax-nul 4977  ax-pow 5029  ax-pr 5090
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2060  df-eu 2633  df-mo 2634  df-clab 2789  df-cleq 2795  df-clel 2798  df-nfc 2933  df-ral 3097  df-rex 3098  df-rab 3101  df-v 3389  df-sbc 3628  df-dif 3766  df-un 3768  df-in 3770  df-ss 3777  df-nul 4111  df-if 4274  df-sn 4365  df-pr 4367  df-op 4371  df-uni 4624  df-br 4838  df-opab 4900  df-mpt 4917  df-id 5213  df-xp 5311  df-rel 5312  df-cnv 5313  df-co 5314  df-dm 5315  df-iota 6058  df-fun 6097  df-fv 6103  df-ats 35041
This theorem is referenced by:  atlatmstc  35093  atlatle  35094  pmapssbaN  35534  pmaple  35535  polsubN  35681  2polvalN  35688  2polssN  35689  3polN  35690  2pmaplubN  35700  paddunN  35701  poldmj1N  35702  pnonsingN  35707  ispsubcl2N  35721  psubclinN  35722  paddatclN  35723  polsubclN  35726  poml4N  35727
  Copyright terms: Public domain W3C validator