| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | lerel 11326 | . . 3
⊢ Rel
≤ | 
| 2 |  | lerelxr 11325 | . . . . . . . . . . 11
⊢  ≤
⊆ (ℝ* × ℝ*) | 
| 3 | 2 | brel 5749 | . . . . . . . . . 10
⊢ (𝑥 ≤ 𝑦 → (𝑥 ∈ ℝ* ∧ 𝑦 ∈
ℝ*)) | 
| 4 | 3 | adantr 480 | . . . . . . . . 9
⊢ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → (𝑥 ∈ ℝ* ∧ 𝑦 ∈
ℝ*)) | 
| 5 | 4 | simpld 494 | . . . . . . . 8
⊢ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ∈ ℝ*) | 
| 6 | 4 | simprd 495 | . . . . . . . 8
⊢ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑦 ∈ ℝ*) | 
| 7 | 2 | brel 5749 | . . . . . . . . . 10
⊢ (𝑦 ≤ 𝑧 → (𝑦 ∈ ℝ* ∧ 𝑧 ∈
ℝ*)) | 
| 8 | 7 | simprd 495 | . . . . . . . . 9
⊢ (𝑦 ≤ 𝑧 → 𝑧 ∈ ℝ*) | 
| 9 | 8 | adantl 481 | . . . . . . . 8
⊢ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑧 ∈ ℝ*) | 
| 10 | 5, 6, 9 | 3jca 1128 | . . . . . . 7
⊢ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → (𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*
∧ 𝑧 ∈
ℝ*)) | 
| 11 |  | xrletr 13201 | . . . . . . 7
⊢ ((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ* ∧ 𝑧
∈ ℝ*) → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧)) | 
| 12 | 10, 11 | mpcom 38 | . . . . . 6
⊢ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧) | 
| 13 | 12 | ax-gen 1794 | . . . . 5
⊢
∀𝑧((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧) | 
| 14 | 13 | gen2 1795 | . . . 4
⊢
∀𝑥∀𝑦∀𝑧((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧) | 
| 15 |  | cotr 6129 | . . . 4
⊢ (( ≤
∘ ≤ ) ⊆ ≤ ↔ ∀𝑥∀𝑦∀𝑧((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧)) | 
| 16 | 14, 15 | mpbir 231 | . . 3
⊢ ( ≤
∘ ≤ ) ⊆ ≤ | 
| 17 |  | asymref 6135 | . . . 4
⊢ (( ≤
∩ ◡ ≤ ) = ( I ↾ ∪ ∪ ≤ ) ↔ ∀𝑥 ∈ ∪ ∪ ≤ ∀𝑦((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) ↔ 𝑥 = 𝑦)) | 
| 18 |  | simpr 484 | . . . . . . . . 9
⊢ ((𝑥 ∈ ℝ*
∧ (𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)) → (𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)) | 
| 19 | 2 | brel 5749 | . . . . . . . . . . . 12
⊢ (𝑦 ≤ 𝑥 → (𝑦 ∈ ℝ* ∧ 𝑥 ∈
ℝ*)) | 
| 20 | 19 | simpld 494 | . . . . . . . . . . 11
⊢ (𝑦 ≤ 𝑥 → 𝑦 ∈ ℝ*) | 
| 21 | 20 | adantl 481 | . . . . . . . . . 10
⊢ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) → 𝑦 ∈ ℝ*) | 
| 22 |  | xrletri3 13197 | . . . . . . . . . 10
⊢ ((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) → (𝑥 = 𝑦 ↔ (𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥))) | 
| 23 | 21, 22 | sylan2 593 | . . . . . . . . 9
⊢ ((𝑥 ∈ ℝ*
∧ (𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)) → (𝑥 = 𝑦 ↔ (𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥))) | 
| 24 | 18, 23 | mpbird 257 | . . . . . . . 8
⊢ ((𝑥 ∈ ℝ*
∧ (𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)) → 𝑥 = 𝑦) | 
| 25 | 24 | ex 412 | . . . . . . 7
⊢ (𝑥 ∈ ℝ*
→ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) → 𝑥 = 𝑦)) | 
| 26 |  | xrleid 13194 | . . . . . . . . 9
⊢ (𝑥 ∈ ℝ*
→ 𝑥 ≤ 𝑥) | 
| 27 | 26, 26 | jca 511 | . . . . . . . 8
⊢ (𝑥 ∈ ℝ*
→ (𝑥 ≤ 𝑥 ∧ 𝑥 ≤ 𝑥)) | 
| 28 |  | breq2 5146 | . . . . . . . . 9
⊢ (𝑥 = 𝑦 → (𝑥 ≤ 𝑥 ↔ 𝑥 ≤ 𝑦)) | 
| 29 |  | breq1 5145 | . . . . . . . . 9
⊢ (𝑥 = 𝑦 → (𝑥 ≤ 𝑥 ↔ 𝑦 ≤ 𝑥)) | 
| 30 | 28, 29 | anbi12d 632 | . . . . . . . 8
⊢ (𝑥 = 𝑦 → ((𝑥 ≤ 𝑥 ∧ 𝑥 ≤ 𝑥) ↔ (𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥))) | 
| 31 | 27, 30 | syl5ibcom 245 | . . . . . . 7
⊢ (𝑥 ∈ ℝ*
→ (𝑥 = 𝑦 → (𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥))) | 
| 32 | 25, 31 | impbid 212 | . . . . . 6
⊢ (𝑥 ∈ ℝ*
→ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) ↔ 𝑥 = 𝑦)) | 
| 33 | 32 | alrimiv 1926 | . . . . 5
⊢ (𝑥 ∈ ℝ*
→ ∀𝑦((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) ↔ 𝑥 = 𝑦)) | 
| 34 |  | lefld 18638 | . . . . . 6
⊢
ℝ* = ∪ ∪ ≤ | 
| 35 | 34 | eqcomi 2745 | . . . . 5
⊢ ∪ ∪ ≤ =
ℝ* | 
| 36 | 33, 35 | eleq2s 2858 | . . . 4
⊢ (𝑥 ∈ ∪ ∪ ≤ → ∀𝑦((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) ↔ 𝑥 = 𝑦)) | 
| 37 | 17, 36 | mprgbir 3067 | . . 3
⊢ ( ≤
∩ ◡ ≤ ) = ( I ↾ ∪ ∪ ≤ ) | 
| 38 |  | xrex 13030 | . . . . . 6
⊢
ℝ* ∈ V | 
| 39 | 38, 38 | xpex 7774 | . . . . 5
⊢
(ℝ* × ℝ*) ∈
V | 
| 40 | 39, 2 | ssexi 5321 | . . . 4
⊢  ≤
∈ V | 
| 41 |  | isps 18614 | . . . 4
⊢ ( ≤
∈ V → ( ≤ ∈ PosetRel ↔ (Rel ≤ ∧ ( ≤ ∘
≤ ) ⊆ ≤ ∧ ( ≤ ∩ ◡ ≤ ) = ( I ↾ ∪ ∪ ≤ )))) | 
| 42 | 40, 41 | ax-mp 5 | . . 3
⊢ ( ≤
∈ PosetRel ↔ (Rel ≤ ∧ ( ≤ ∘ ≤ ) ⊆ ≤ ∧
( ≤ ∩ ◡ ≤ ) = ( I ↾
∪ ∪ ≤
))) | 
| 43 | 1, 16, 37, 42 | mpbir3an 1341 | . 2
⊢  ≤
∈ PosetRel | 
| 44 |  | xrletri 13196 | . . . 4
⊢ ((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) → (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥)) | 
| 45 | 44 | rgen2 3198 | . . 3
⊢
∀𝑥 ∈
ℝ* ∀𝑦 ∈ ℝ* (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥) | 
| 46 |  | qfto 6140 | . . 3
⊢
((ℝ* × ℝ*) ⊆ ( ≤ ∪
◡ ≤ ) ↔ ∀𝑥 ∈ ℝ*
∀𝑦 ∈
ℝ* (𝑥 ≤
𝑦 ∨ 𝑦 ≤ 𝑥)) | 
| 47 | 45, 46 | mpbir 231 | . 2
⊢
(ℝ* × ℝ*) ⊆ ( ≤ ∪
◡ ≤ ) | 
| 48 |  | ledm 18636 | . . 3
⊢
ℝ* = dom ≤ | 
| 49 | 48 | istsr 18629 | . 2
⊢ ( ≤
∈ TosetRel ↔ ( ≤ ∈ PosetRel ∧ (ℝ*
× ℝ*) ⊆ ( ≤ ∪ ◡ ≤ ))) | 
| 50 | 43, 47, 49 | mpbir2an 711 | 1
⊢  ≤
∈ TosetRel |