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

Theorem atssbase 29480
Description: The set of atoms is a subset of the base set. (atssch 22923 analog.) (Contributed by NM, 21-Oct-2011.)
Hypotheses
Ref Expression
atombase.b  |-  B  =  ( Base `  K
)
atombase.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
atssbase  |-  A  C_  B

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 29479 . 2  |-  ( x  e.  A  ->  x  e.  B )
43ssriv 3184 1  |-  A  C_  B
Colors of variables: wff set class
Syntax hints:    = wceq 1623    C_ wss 3152   ` cfv 5255   Basecbs 13148   Atomscatm 29453
This theorem is referenced by:  atlatmstc  29509  atlatle  29510  pmapssbaN  29949  pmaple  29950  polsubN  30096  2polvalN  30103  2polssN  30104  3polN  30105  2pmaplubN  30115  paddunN  30116  poldmj1N  30117  pnonsingN  30122  ispsubcl2N  30136  psubclinN  30137  paddatclN  30138  polsubclN  30141  poml4N  30142
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ats 29457
  Copyright terms: Public domain W3C validator