Metamath Proof Explorer
Description: Relationship between <_ and >_ using hypotheses. (Contributed by David A. Wheeler, 10-May-2015) (New usage is discouraged.)
|
|
Ref |
Expression |
|
Hypotheses |
gte-lteh.1 |
⊢ 𝐴 ∈ V |
|
|
gte-lteh.2 |
⊢ 𝐵 ∈ V |
|
Assertion |
gte-lteh |
⊢ ( 𝐴 ≥ 𝐵 ↔ 𝐵 ≤ 𝐴 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
gte-lteh.1 |
⊢ 𝐴 ∈ V |
2 |
|
gte-lteh.2 |
⊢ 𝐵 ∈ V |
3 |
|
df-gte |
⊢ ≥ = ◡ ≤ |
4 |
3
|
breqi |
⊢ ( 𝐴 ≥ 𝐵 ↔ 𝐴 ◡ ≤ 𝐵 ) |
5 |
1 2
|
brcnv |
⊢ ( 𝐴 ◡ ≤ 𝐵 ↔ 𝐵 ≤ 𝐴 ) |
6 |
4 5
|
bitri |
⊢ ( 𝐴 ≥ 𝐵 ↔ 𝐵 ≤ 𝐴 ) |