This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem lidl0

Description: Every ring contains a zero ideal. (Contributed by Stefan O'Rear, 3-Jan-2015) (Proof shortened by AV, 18-Apr-2025)

Ref Expression
Hypotheses rnglidl0.u ⊢ 𝑈 = ( LIdeal ‘ 𝑅 )
rnglidl0.z ⊢ 0 = ( 0g ‘ 𝑅 )
Assertion lidl0 ( 𝑅 ∈ Ring → { 0 } ∈ 𝑈 )

Proof

Step Hyp Ref Expression
1 rnglidl0.u ⊢ 𝑈 = ( LIdeal ‘ 𝑅 )
2 rnglidl0.z ⊢ 0 = ( 0g ‘ 𝑅 )
3 ringrng ⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ Rng )
4 1 2 rnglidl0 ⊢ ( 𝑅 ∈ Rng → { 0 } ∈ 𝑈 )
5 3 4 syl ⊢ ( 𝑅 ∈ Ring → { 0 } ∈ 𝑈 )