Theorem atssbase 36420

Description: The set of atoms is a subset of the base set. (atssch 30114 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 ⊆ 𝐵

Proof of Theorem atssbase

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		atombase.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
2		atombase.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
3	1, 2	atbase 36419	. 2 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)
4	3	ssriv 3971	1 ⊢ 𝐴 ⊆ 𝐵

Syntax hints:   = wceq 1533   ⊆ wss 3936  ‘cfv 6350  Basecbs 16477  Atomscatm 36393

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-iota 6309  df-fun 6352  df-fv 6358  df-ats 36397

This theorem is referenced by:  atlatmstc  36449  atlatle  36450  pmapssbaN  36890  pmaple  36891  polsubN  37037  2polvalN  37044  2polssN  37045  3polN  37046  2pmaplubN  37056  paddunN  37057  poldmj1N  37058  pnonsingN  37063  ispsubcl2N  37077  psubclinN  37078  paddatclN  37079  polsubclN  37082  poml4N  37083