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Theorem atssbase 30150
Description: The set of atoms is a subset of the base set. (atssch 23848 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 30149 . 2  |-  ( x  e.  A  ->  x  e.  B )
43ssriv 3354 1  |-  A  C_  B
Colors of variables: wff set class
Syntax hints:    = wceq 1653    C_ wss 3322   ` cfv 5456   Basecbs 13471   Atomscatm 30123
This theorem is referenced by:  atlatmstc  30179  atlatle  30180  pmapssbaN  30619  pmaple  30620  polsubN  30766  2polvalN  30773  2polssN  30774  3polN  30775  2pmaplubN  30785  paddunN  30786  poldmj1N  30787  pnonsingN  30792  ispsubcl2N  30806  psubclinN  30807  paddatclN  30808  polsubclN  30811  poml4N  30812
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fv 5464  df-ats 30127
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