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

Theorem atssbase 30102
Description: The set of atoms is a subset of the base set. (atssch 22939 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 30101 . 2  |-  ( x  e.  A  ->  x  e.  B )
43ssriv 3197 1  |-  A  C_  B
Colors of variables: wff set class
Syntax hints:    = wceq 1632    C_ wss 3165   ` cfv 5271   Basecbs 13164   Atomscatm 30075
This theorem is referenced by:  atlatmstc  30131  atlatle  30132  pmapssbaN  30571  pmaple  30572  polsubN  30718  2polvalN  30725  2polssN  30726  3polN  30727  2pmaplubN  30737  paddunN  30738  poldmj1N  30739  pnonsingN  30744  ispsubcl2N  30758  psubclinN  30759  paddatclN  30760  polsubclN  30763  poml4N  30764
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ats 30079
  Copyright terms: Public domain W3C validator