| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | simpr 484 | . . . . . . 7
⊢ (((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) ∧ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) = (𝑌‘2))) → ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) = (𝑌‘2))) | 
| 2 |  | rrx2pnecoorneor.i | . . . . . . . . 9
⊢ 𝐼 = {1, 2} | 
| 3 | 2 | raleqi 3323 | . . . . . . . 8
⊢
(∀𝑖 ∈
𝐼 (𝑋‘𝑖) = (𝑌‘𝑖) ↔ ∀𝑖 ∈ {1, 2} (𝑋‘𝑖) = (𝑌‘𝑖)) | 
| 4 |  | 1ex 11258 | . . . . . . . . 9
⊢ 1 ∈
V | 
| 5 |  | 2ex 12344 | . . . . . . . . 9
⊢ 2 ∈
V | 
| 6 |  | fveq2 6905 | . . . . . . . . . 10
⊢ (𝑖 = 1 → (𝑋‘𝑖) = (𝑋‘1)) | 
| 7 |  | fveq2 6905 | . . . . . . . . . 10
⊢ (𝑖 = 1 → (𝑌‘𝑖) = (𝑌‘1)) | 
| 8 | 6, 7 | eqeq12d 2752 | . . . . . . . . 9
⊢ (𝑖 = 1 → ((𝑋‘𝑖) = (𝑌‘𝑖) ↔ (𝑋‘1) = (𝑌‘1))) | 
| 9 |  | fveq2 6905 | . . . . . . . . . 10
⊢ (𝑖 = 2 → (𝑋‘𝑖) = (𝑋‘2)) | 
| 10 |  | fveq2 6905 | . . . . . . . . . 10
⊢ (𝑖 = 2 → (𝑌‘𝑖) = (𝑌‘2)) | 
| 11 | 9, 10 | eqeq12d 2752 | . . . . . . . . 9
⊢ (𝑖 = 2 → ((𝑋‘𝑖) = (𝑌‘𝑖) ↔ (𝑋‘2) = (𝑌‘2))) | 
| 12 | 4, 5, 8, 11 | ralpr 4699 | . . . . . . . 8
⊢
(∀𝑖 ∈
{1, 2} (𝑋‘𝑖) = (𝑌‘𝑖) ↔ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) = (𝑌‘2))) | 
| 13 | 3, 12 | bitri 275 | . . . . . . 7
⊢
(∀𝑖 ∈
𝐼 (𝑋‘𝑖) = (𝑌‘𝑖) ↔ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) = (𝑌‘2))) | 
| 14 | 1, 13 | sylibr 234 | . . . . . 6
⊢ (((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) ∧ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) = (𝑌‘2))) → ∀𝑖 ∈ 𝐼 (𝑋‘𝑖) = (𝑌‘𝑖)) | 
| 15 |  | elmapfn 8906 | . . . . . . . . . 10
⊢ (𝑋 ∈ (ℝ
↑m 𝐼)
→ 𝑋 Fn 𝐼) | 
| 16 |  | rrx2pnecoorneor.b | . . . . . . . . . 10
⊢ 𝑃 = (ℝ ↑m
𝐼) | 
| 17 | 15, 16 | eleq2s 2858 | . . . . . . . . 9
⊢ (𝑋 ∈ 𝑃 → 𝑋 Fn 𝐼) | 
| 18 |  | elmapfn 8906 | . . . . . . . . . 10
⊢ (𝑌 ∈ (ℝ
↑m 𝐼)
→ 𝑌 Fn 𝐼) | 
| 19 | 18, 16 | eleq2s 2858 | . . . . . . . . 9
⊢ (𝑌 ∈ 𝑃 → 𝑌 Fn 𝐼) | 
| 20 | 17, 19 | anim12i 613 | . . . . . . . 8
⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → (𝑋 Fn 𝐼 ∧ 𝑌 Fn 𝐼)) | 
| 21 | 20 | adantr 480 | . . . . . . 7
⊢ (((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) ∧ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) = (𝑌‘2))) → (𝑋 Fn 𝐼 ∧ 𝑌 Fn 𝐼)) | 
| 22 |  | eqfnfv 7050 | . . . . . . 7
⊢ ((𝑋 Fn 𝐼 ∧ 𝑌 Fn 𝐼) → (𝑋 = 𝑌 ↔ ∀𝑖 ∈ 𝐼 (𝑋‘𝑖) = (𝑌‘𝑖))) | 
| 23 | 21, 22 | syl 17 | . . . . . 6
⊢ (((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) ∧ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) = (𝑌‘2))) → (𝑋 = 𝑌 ↔ ∀𝑖 ∈ 𝐼 (𝑋‘𝑖) = (𝑌‘𝑖))) | 
| 24 | 14, 23 | mpbird 257 | . . . . 5
⊢ (((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) ∧ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) = (𝑌‘2))) → 𝑋 = 𝑌) | 
| 25 | 24 | ex 412 | . . . 4
⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → (((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) = (𝑌‘2)) → 𝑋 = 𝑌)) | 
| 26 | 25 | necon3ad 2952 | . . 3
⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → (𝑋 ≠ 𝑌 → ¬ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) = (𝑌‘2)))) | 
| 27 | 26 | 3impia 1117 | . 2
⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → ¬ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) = (𝑌‘2))) | 
| 28 |  | neorian 3036 | . 2
⊢ (((𝑋‘1) ≠ (𝑌‘1) ∨ (𝑋‘2) ≠ (𝑌‘2)) ↔ ¬ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) = (𝑌‘2))) | 
| 29 | 27, 28 | sylibr 234 | 1
⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → ((𝑋‘1) ≠ (𝑌‘1) ∨ (𝑋‘2) ≠ (𝑌‘2))) |