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Theorem subrngss 20565
Description: A subring is a subset. (Contributed by AV, 14-Feb-2025.)
Hypothesis
Ref Expression
subrngss.1 𝐵 = (Base‘𝑅)
Assertion
Ref Expression
subrngss (𝐴 ∈ (SubRng‘𝑅) → 𝐴𝐵)

Proof of Theorem subrngss
StepHypRef Expression
1 subrngss.1 . . 3 𝐵 = (Base‘𝑅)
21issubrng 20564 . 2 (𝐴 ∈ (SubRng‘𝑅) ↔ (𝑅 ∈ Rng ∧ (𝑅s 𝐴) ∈ Rng ∧ 𝐴𝐵))
32simp3bi 1146 1 (𝐴 ∈ (SubRng‘𝑅) → 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  wss 3963  cfv 6563  (class class class)co 7431  Basecbs 17245  s cress 17274  Rngcrng 20170  SubRngcsubrng 20562
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-subrng 20563
This theorem is referenced by:  subrngsubg  20569  subrngmre  20579  subsubrng  20580  rhmimasubrnglem  20582  rhmimasubrng  20583
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