Step |
Hyp |
Ref |
Expression |
1 |
|
rrx2plord.o |
⊢ 𝑂 = { 〈 𝑥 , 𝑦 〉 ∣ ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅 ) ∧ ( ( 𝑥 ‘ 1 ) < ( 𝑦 ‘ 1 ) ∨ ( ( 𝑥 ‘ 1 ) = ( 𝑦 ‘ 1 ) ∧ ( 𝑥 ‘ 2 ) < ( 𝑦 ‘ 2 ) ) ) ) } |
2 |
|
rrx2plord2.r |
⊢ 𝑅 = ( ℝ ↑m { 1 , 2 } ) |
3 |
1
|
rrx2plord |
⊢ ( ( 𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅 ) → ( 𝑋 𝑂 𝑌 ↔ ( ( 𝑋 ‘ 1 ) < ( 𝑌 ‘ 1 ) ∨ ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋 ‘ 2 ) < ( 𝑌 ‘ 2 ) ) ) ) ) |
4 |
3
|
3adant3 |
⊢ ( ( 𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅 ∧ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) → ( 𝑋 𝑂 𝑌 ↔ ( ( 𝑋 ‘ 1 ) < ( 𝑌 ‘ 1 ) ∨ ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋 ‘ 2 ) < ( 𝑌 ‘ 2 ) ) ) ) ) |
5 |
|
eqid |
⊢ { 1 , 2 } = { 1 , 2 } |
6 |
5 2
|
rrx2pxel |
⊢ ( 𝑋 ∈ 𝑅 → ( 𝑋 ‘ 1 ) ∈ ℝ ) |
7 |
6
|
adantr |
⊢ ( ( 𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅 ) → ( 𝑋 ‘ 1 ) ∈ ℝ ) |
8 |
|
ltne |
⊢ ( ( ( 𝑋 ‘ 1 ) ∈ ℝ ∧ ( 𝑋 ‘ 1 ) < ( 𝑌 ‘ 1 ) ) → ( 𝑌 ‘ 1 ) ≠ ( 𝑋 ‘ 1 ) ) |
9 |
8
|
necomd |
⊢ ( ( ( 𝑋 ‘ 1 ) ∈ ℝ ∧ ( 𝑋 ‘ 1 ) < ( 𝑌 ‘ 1 ) ) → ( 𝑋 ‘ 1 ) ≠ ( 𝑌 ‘ 1 ) ) |
10 |
7 9
|
sylan |
⊢ ( ( ( 𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅 ) ∧ ( 𝑋 ‘ 1 ) < ( 𝑌 ‘ 1 ) ) → ( 𝑋 ‘ 1 ) ≠ ( 𝑌 ‘ 1 ) ) |
11 |
10
|
ex |
⊢ ( ( 𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅 ) → ( ( 𝑋 ‘ 1 ) < ( 𝑌 ‘ 1 ) → ( 𝑋 ‘ 1 ) ≠ ( 𝑌 ‘ 1 ) ) ) |
12 |
|
eqneqall |
⊢ ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) → ( ( 𝑋 ‘ 1 ) ≠ ( 𝑌 ‘ 1 ) → ( 𝑋 ‘ 2 ) < ( 𝑌 ‘ 2 ) ) ) |
13 |
11 12
|
syl9 |
⊢ ( ( 𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅 ) → ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) → ( ( 𝑋 ‘ 1 ) < ( 𝑌 ‘ 1 ) → ( 𝑋 ‘ 2 ) < ( 𝑌 ‘ 2 ) ) ) ) |
14 |
13
|
3impia |
⊢ ( ( 𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅 ∧ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) → ( ( 𝑋 ‘ 1 ) < ( 𝑌 ‘ 1 ) → ( 𝑋 ‘ 2 ) < ( 𝑌 ‘ 2 ) ) ) |
15 |
14
|
com12 |
⊢ ( ( 𝑋 ‘ 1 ) < ( 𝑌 ‘ 1 ) → ( ( 𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅 ∧ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) → ( 𝑋 ‘ 2 ) < ( 𝑌 ‘ 2 ) ) ) |
16 |
|
simpr |
⊢ ( ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋 ‘ 2 ) < ( 𝑌 ‘ 2 ) ) → ( 𝑋 ‘ 2 ) < ( 𝑌 ‘ 2 ) ) |
17 |
16
|
a1d |
⊢ ( ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋 ‘ 2 ) < ( 𝑌 ‘ 2 ) ) → ( ( 𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅 ∧ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) → ( 𝑋 ‘ 2 ) < ( 𝑌 ‘ 2 ) ) ) |
18 |
15 17
|
jaoi |
⊢ ( ( ( 𝑋 ‘ 1 ) < ( 𝑌 ‘ 1 ) ∨ ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋 ‘ 2 ) < ( 𝑌 ‘ 2 ) ) ) → ( ( 𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅 ∧ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) → ( 𝑋 ‘ 2 ) < ( 𝑌 ‘ 2 ) ) ) |
19 |
18
|
com12 |
⊢ ( ( 𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅 ∧ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) → ( ( ( 𝑋 ‘ 1 ) < ( 𝑌 ‘ 1 ) ∨ ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋 ‘ 2 ) < ( 𝑌 ‘ 2 ) ) ) → ( 𝑋 ‘ 2 ) < ( 𝑌 ‘ 2 ) ) ) |
20 |
|
olc |
⊢ ( ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋 ‘ 2 ) < ( 𝑌 ‘ 2 ) ) → ( ( 𝑋 ‘ 1 ) < ( 𝑌 ‘ 1 ) ∨ ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋 ‘ 2 ) < ( 𝑌 ‘ 2 ) ) ) ) |
21 |
20
|
ex |
⊢ ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) → ( ( 𝑋 ‘ 2 ) < ( 𝑌 ‘ 2 ) → ( ( 𝑋 ‘ 1 ) < ( 𝑌 ‘ 1 ) ∨ ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋 ‘ 2 ) < ( 𝑌 ‘ 2 ) ) ) ) ) |
22 |
21
|
3ad2ant3 |
⊢ ( ( 𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅 ∧ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) → ( ( 𝑋 ‘ 2 ) < ( 𝑌 ‘ 2 ) → ( ( 𝑋 ‘ 1 ) < ( 𝑌 ‘ 1 ) ∨ ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋 ‘ 2 ) < ( 𝑌 ‘ 2 ) ) ) ) ) |
23 |
19 22
|
impbid |
⊢ ( ( 𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅 ∧ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) → ( ( ( 𝑋 ‘ 1 ) < ( 𝑌 ‘ 1 ) ∨ ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋 ‘ 2 ) < ( 𝑌 ‘ 2 ) ) ) ↔ ( 𝑋 ‘ 2 ) < ( 𝑌 ‘ 2 ) ) ) |
24 |
4 23
|
bitrd |
⊢ ( ( 𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅 ∧ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) → ( 𝑋 𝑂 𝑌 ↔ ( 𝑋 ‘ 2 ) < ( 𝑌 ‘ 2 ) ) ) |