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Theorem atssbase 38160
Description: The set of atoms is a subset of the base set. (atssch 31596 analog.) (Contributed by NM, 21-Oct-2011.)
Hypotheses
Ref Expression
atombase.b 𝐡 = (Baseβ€˜πΎ)
atombase.a 𝐴 = (Atomsβ€˜πΎ)
Assertion
Ref Expression
atssbase 𝐴 βŠ† 𝐡

Proof of Theorem atssbase
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 atombase.b . . 3 𝐡 = (Baseβ€˜πΎ)
2 atombase.a . . 3 𝐴 = (Atomsβ€˜πΎ)
31, 2atbase 38159 . 2 (π‘₯ ∈ 𝐴 β†’ π‘₯ ∈ 𝐡)
43ssriv 3987 1 𝐴 βŠ† 𝐡
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542   βŠ† wss 3949  β€˜cfv 6544  Basecbs 17144  Atomscatm 38133
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-ats 38137
This theorem is referenced by:  atlatmstc  38189  atlatle  38190  pmapssbaN  38631  pmaple  38632  polsubN  38778  2polvalN  38785  2polssN  38786  3polN  38787  2pmaplubN  38797  paddunN  38798  poldmj1N  38799  pnonsingN  38804  ispsubcl2N  38818  psubclinN  38819  paddatclN  38820  polsubclN  38823  poml4N  38824
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