Step |
Hyp |
Ref |
Expression |
1 |
|
tglineelsb2.p |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
tglineelsb2.i |
⊢ 𝐼 = ( Itv ‘ 𝐺 ) |
3 |
|
tglineelsb2.l |
⊢ 𝐿 = ( LineG ‘ 𝐺 ) |
4 |
|
tglineelsb2.g |
⊢ ( 𝜑 → 𝐺 ∈ TarskiG ) |
5 |
|
tgelrnln.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐵 ) |
6 |
|
tgelrnln.y |
⊢ ( 𝜑 → 𝑌 ∈ 𝐵 ) |
7 |
|
tgelrnln.d |
⊢ ( 𝜑 → 𝑋 ≠ 𝑌 ) |
8 |
|
df-ov |
⊢ ( 𝑋 𝐿 𝑌 ) = ( 𝐿 ‘ 〈 𝑋 , 𝑌 〉 ) |
9 |
1 3 2
|
tglnfn |
⊢ ( 𝐺 ∈ TarskiG → 𝐿 Fn ( ( 𝐵 × 𝐵 ) ∖ I ) ) |
10 |
4 9
|
syl |
⊢ ( 𝜑 → 𝐿 Fn ( ( 𝐵 × 𝐵 ) ∖ I ) ) |
11 |
5 6
|
opelxpd |
⊢ ( 𝜑 → 〈 𝑋 , 𝑌 〉 ∈ ( 𝐵 × 𝐵 ) ) |
12 |
|
df-br |
⊢ ( 𝑋 I 𝑌 ↔ 〈 𝑋 , 𝑌 〉 ∈ I ) |
13 |
|
ideqg |
⊢ ( 𝑌 ∈ 𝐵 → ( 𝑋 I 𝑌 ↔ 𝑋 = 𝑌 ) ) |
14 |
12 13
|
bitr3id |
⊢ ( 𝑌 ∈ 𝐵 → ( 〈 𝑋 , 𝑌 〉 ∈ I ↔ 𝑋 = 𝑌 ) ) |
15 |
14
|
necon3bbid |
⊢ ( 𝑌 ∈ 𝐵 → ( ¬ 〈 𝑋 , 𝑌 〉 ∈ I ↔ 𝑋 ≠ 𝑌 ) ) |
16 |
15
|
biimpar |
⊢ ( ( 𝑌 ∈ 𝐵 ∧ 𝑋 ≠ 𝑌 ) → ¬ 〈 𝑋 , 𝑌 〉 ∈ I ) |
17 |
6 7 16
|
syl2anc |
⊢ ( 𝜑 → ¬ 〈 𝑋 , 𝑌 〉 ∈ I ) |
18 |
11 17
|
eldifd |
⊢ ( 𝜑 → 〈 𝑋 , 𝑌 〉 ∈ ( ( 𝐵 × 𝐵 ) ∖ I ) ) |
19 |
|
fnfvelrn |
⊢ ( ( 𝐿 Fn ( ( 𝐵 × 𝐵 ) ∖ I ) ∧ 〈 𝑋 , 𝑌 〉 ∈ ( ( 𝐵 × 𝐵 ) ∖ I ) ) → ( 𝐿 ‘ 〈 𝑋 , 𝑌 〉 ) ∈ ran 𝐿 ) |
20 |
10 18 19
|
syl2anc |
⊢ ( 𝜑 → ( 𝐿 ‘ 〈 𝑋 , 𝑌 〉 ) ∈ ran 𝐿 ) |
21 |
8 20
|
eqeltrid |
⊢ ( 𝜑 → ( 𝑋 𝐿 𝑌 ) ∈ ran 𝐿 ) |