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Theorem atssbase 29773
Description: The set of atoms is a subset of the base set. (atssch 23799 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 29772 . 2  |-  ( x  e.  A  ->  x  e.  B )
43ssriv 3312 1  |-  A  C_  B
Colors of variables: wff set class
Syntax hints:    = wceq 1649    C_ wss 3280   ` cfv 5413   Basecbs 13424   Atomscatm 29746
This theorem is referenced by:  atlatmstc  29802  atlatle  29803  pmapssbaN  30242  pmaple  30243  polsubN  30389  2polvalN  30396  2polssN  30397  3polN  30398  2pmaplubN  30408  paddunN  30409  poldmj1N  30410  pnonsingN  30415  ispsubcl2N  30429  psubclinN  30430  paddatclN  30431  polsubclN  30434  poml4N  30435
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fv 5421  df-ats 29750
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