Step |
Hyp |
Ref |
Expression |
1 |
|
vr1val.1 |
⊢ 𝑋 = ( var1 ‘ 𝑅 ) |
2 |
|
vr1cl2.2 |
⊢ 𝑆 = ( PwSer1 ‘ 𝑅 ) |
3 |
|
vr1cl2.3 |
⊢ 𝐵 = ( Base ‘ 𝑆 ) |
4 |
1
|
vr1val |
⊢ 𝑋 = ( ( 1o mVar 𝑅 ) ‘ ∅ ) |
5 |
|
eqid |
⊢ ( 1o mPwSer 𝑅 ) = ( 1o mPwSer 𝑅 ) |
6 |
|
eqid |
⊢ ( 1o mVar 𝑅 ) = ( 1o mVar 𝑅 ) |
7 |
|
eqid |
⊢ ( Base ‘ ( 1o mPwSer 𝑅 ) ) = ( Base ‘ ( 1o mPwSer 𝑅 ) ) |
8 |
|
1on |
⊢ 1o ∈ On |
9 |
8
|
a1i |
⊢ ( 𝑅 ∈ Ring → 1o ∈ On ) |
10 |
|
id |
⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ Ring ) |
11 |
|
0lt1o |
⊢ ∅ ∈ 1o |
12 |
11
|
a1i |
⊢ ( 𝑅 ∈ Ring → ∅ ∈ 1o ) |
13 |
5 6 7 9 10 12
|
mvrcl2 |
⊢ ( 𝑅 ∈ Ring → ( ( 1o mVar 𝑅 ) ‘ ∅ ) ∈ ( Base ‘ ( 1o mPwSer 𝑅 ) ) ) |
14 |
2
|
psr1val |
⊢ 𝑆 = ( ( 1o ordPwSer 𝑅 ) ‘ ∅ ) |
15 |
|
0ss |
⊢ ∅ ⊆ ( 1o × 1o ) |
16 |
15
|
a1i |
⊢ ( 𝑅 ∈ Ring → ∅ ⊆ ( 1o × 1o ) ) |
17 |
5 14 16
|
opsrbas |
⊢ ( 𝑅 ∈ Ring → ( Base ‘ ( 1o mPwSer 𝑅 ) ) = ( Base ‘ 𝑆 ) ) |
18 |
17 3
|
eqtr4di |
⊢ ( 𝑅 ∈ Ring → ( Base ‘ ( 1o mPwSer 𝑅 ) ) = 𝐵 ) |
19 |
13 18
|
eleqtrd |
⊢ ( 𝑅 ∈ Ring → ( ( 1o mVar 𝑅 ) ‘ ∅ ) ∈ 𝐵 ) |
20 |
4 19
|
eqeltrid |
⊢ ( 𝑅 ∈ Ring → 𝑋 ∈ 𝐵 ) |