Metamath Proof Explorer
Description: Any element is less than or equal to a poset's upper bound (if defined).
(Contributed by NM, 22-Oct-2011) (Revised by NM, 13-Sep-2018)
|
|
Ref |
Expression |
|
Hypotheses |
ple1.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
|
|
ple1.u |
⊢ 𝑈 = ( lub ‘ 𝐾 ) |
|
|
ple1.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
|
|
ple1.1 |
⊢ 1 = ( 1. ‘ 𝐾 ) |
|
|
ple1.k |
⊢ ( 𝜑 → 𝐾 ∈ 𝑉 ) |
|
|
ple1.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐵 ) |
|
|
ple1.d |
⊢ ( 𝜑 → 𝐵 ∈ dom 𝑈 ) |
|
Assertion |
ple1 |
⊢ ( 𝜑 → 𝑋 ≤ 1 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
ple1.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
ple1.u |
⊢ 𝑈 = ( lub ‘ 𝐾 ) |
3 |
|
ple1.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
4 |
|
ple1.1 |
⊢ 1 = ( 1. ‘ 𝐾 ) |
5 |
|
ple1.k |
⊢ ( 𝜑 → 𝐾 ∈ 𝑉 ) |
6 |
|
ple1.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐵 ) |
7 |
|
ple1.d |
⊢ ( 𝜑 → 𝐵 ∈ dom 𝑈 ) |
8 |
1 3 2 5 7 6
|
luble |
⊢ ( 𝜑 → 𝑋 ≤ ( 𝑈 ‘ 𝐵 ) ) |
9 |
1 2 4
|
p1val |
⊢ ( 𝐾 ∈ 𝑉 → 1 = ( 𝑈 ‘ 𝐵 ) ) |
10 |
5 9
|
syl |
⊢ ( 𝜑 → 1 = ( 𝑈 ‘ 𝐵 ) ) |
11 |
8 10
|
breqtrrd |
⊢ ( 𝜑 → 𝑋 ≤ 1 ) |