Step |
Hyp |
Ref |
Expression |
1 |
|
mxidlval.1 |
⊢ 𝐵 = ( Base ‘ 𝑅 ) |
2 |
|
mxidln1.1 |
⊢ 1 = ( 1r ‘ 𝑅 ) |
3 |
1
|
mxidlnr |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) → 𝑀 ≠ 𝐵 ) |
4 |
1
|
mxidlidl |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) → 𝑀 ∈ ( LIdeal ‘ 𝑅 ) ) |
5 |
|
eqid |
⊢ ( LIdeal ‘ 𝑅 ) = ( LIdeal ‘ 𝑅 ) |
6 |
5 1 2
|
lidl1el |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑀 ∈ ( LIdeal ‘ 𝑅 ) ) → ( 1 ∈ 𝑀 ↔ 𝑀 = 𝐵 ) ) |
7 |
4 6
|
syldan |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) → ( 1 ∈ 𝑀 ↔ 𝑀 = 𝐵 ) ) |
8 |
7
|
necon3bbid |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) → ( ¬ 1 ∈ 𝑀 ↔ 𝑀 ≠ 𝐵 ) ) |
9 |
3 8
|
mpbird |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) → ¬ 1 ∈ 𝑀 ) |