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Theorem subrngss 20492
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 20491 . 2 (𝐴 ∈ (SubRng‘𝑅) ↔ (𝑅 ∈ Rng ∧ (𝑅s 𝐴) ∈ Rng ∧ 𝐴𝐵))
32simp3bi 1144 1 (𝐴 ∈ (SubRng‘𝑅) → 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  wss 3949  cfv 6553  (class class class)co 7426  Basecbs 17187  s cress 17216  Rngcrng 20099  SubRngcsubrng 20489
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fv 6561  df-ov 7429  df-subrng 20490
This theorem is referenced by:  subrngsubg  20496  subrngmre  20506  subsubrng  20507  rhmimasubrnglem  20509  rhmimasubrng  20510
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