Proof of Theorem tsrlemax
Step | Hyp | Ref
| Expression |
1 | | breq2 5044 |
. . 3
⊢ (𝐶 = if(𝐵𝑅𝐶, 𝐶, 𝐵) → (𝐴𝑅𝐶 ↔ 𝐴𝑅if(𝐵𝑅𝐶, 𝐶, 𝐵))) |
2 | 1 | bibi1d 347 |
. 2
⊢ (𝐶 = if(𝐵𝑅𝐶, 𝐶, 𝐵) → ((𝐴𝑅𝐶 ↔ (𝐴𝑅𝐵 ∨ 𝐴𝑅𝐶)) ↔ (𝐴𝑅if(𝐵𝑅𝐶, 𝐶, 𝐵) ↔ (𝐴𝑅𝐵 ∨ 𝐴𝑅𝐶)))) |
3 | | breq2 5044 |
. . 3
⊢ (𝐵 = if(𝐵𝑅𝐶, 𝐶, 𝐵) → (𝐴𝑅𝐵 ↔ 𝐴𝑅if(𝐵𝑅𝐶, 𝐶, 𝐵))) |
4 | 3 | bibi1d 347 |
. 2
⊢ (𝐵 = if(𝐵𝑅𝐶, 𝐶, 𝐵) → ((𝐴𝑅𝐵 ↔ (𝐴𝑅𝐵 ∨ 𝐴𝑅𝐶)) ↔ (𝐴𝑅if(𝐵𝑅𝐶, 𝐶, 𝐵) ↔ (𝐴𝑅𝐵 ∨ 𝐴𝑅𝐶)))) |
5 | | olc 867 |
. . 3
⊢ (𝐴𝑅𝐶 → (𝐴𝑅𝐵 ∨ 𝐴𝑅𝐶)) |
6 | | eqid 2739 |
. . . . . . . . . 10
⊢ dom 𝑅 = dom 𝑅 |
7 | 6 | istsr 17955 |
. . . . . . . . 9
⊢ (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ (dom
𝑅 × dom 𝑅) ⊆ (𝑅 ∪ ◡𝑅))) |
8 | 7 | simplbi 501 |
. . . . . . . 8
⊢ (𝑅 ∈ TosetRel → 𝑅 ∈
PosetRel) |
9 | | pstr 17949 |
. . . . . . . . 9
⊢ ((𝑅 ∈ PosetRel ∧ 𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶) |
10 | 9 | 3expib 1123 |
. . . . . . . 8
⊢ (𝑅 ∈ PosetRel → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)) |
11 | 8, 10 | syl 17 |
. . . . . . 7
⊢ (𝑅 ∈ TosetRel → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)) |
12 | 11 | adantr 484 |
. . . . . 6
⊢ ((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)) |
13 | 12 | expdimp 456 |
. . . . 5
⊢ (((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐴𝑅𝐵) → (𝐵𝑅𝐶 → 𝐴𝑅𝐶)) |
14 | 13 | impancom 455 |
. . . 4
⊢ (((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐵𝑅𝐶) → (𝐴𝑅𝐵 → 𝐴𝑅𝐶)) |
15 | | idd 24 |
. . . 4
⊢ (((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐵𝑅𝐶) → (𝐴𝑅𝐶 → 𝐴𝑅𝐶)) |
16 | 14, 15 | jaod 858 |
. . 3
⊢ (((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐵𝑅𝐶) → ((𝐴𝑅𝐵 ∨ 𝐴𝑅𝐶) → 𝐴𝑅𝐶)) |
17 | 5, 16 | impbid2 229 |
. 2
⊢ (((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐵𝑅𝐶) → (𝐴𝑅𝐶 ↔ (𝐴𝑅𝐵 ∨ 𝐴𝑅𝐶))) |
18 | | orc 866 |
. . 3
⊢ (𝐴𝑅𝐵 → (𝐴𝑅𝐵 ∨ 𝐴𝑅𝐶)) |
19 | | idd 24 |
. . . 4
⊢ (((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ ¬ 𝐵𝑅𝐶) → (𝐴𝑅𝐵 → 𝐴𝑅𝐵)) |
20 | | istsr.1 |
. . . . . . . 8
⊢ 𝑋 = dom 𝑅 |
21 | 20 | tsrlin 17957 |
. . . . . . 7
⊢ ((𝑅 ∈ TosetRel ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵)) |
22 | 21 | 3adant3r1 1183 |
. . . . . 6
⊢ ((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵)) |
23 | 22 | orcanai 1002 |
. . . . 5
⊢ (((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ ¬ 𝐵𝑅𝐶) → 𝐶𝑅𝐵) |
24 | | pstr 17949 |
. . . . . . . . . 10
⊢ ((𝑅 ∈ PosetRel ∧ 𝐴𝑅𝐶 ∧ 𝐶𝑅𝐵) → 𝐴𝑅𝐵) |
25 | 24 | 3expib 1123 |
. . . . . . . . 9
⊢ (𝑅 ∈ PosetRel → ((𝐴𝑅𝐶 ∧ 𝐶𝑅𝐵) → 𝐴𝑅𝐵)) |
26 | 8, 25 | syl 17 |
. . . . . . . 8
⊢ (𝑅 ∈ TosetRel → ((𝐴𝑅𝐶 ∧ 𝐶𝑅𝐵) → 𝐴𝑅𝐵)) |
27 | 26 | adantr 484 |
. . . . . . 7
⊢ ((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝑅𝐶 ∧ 𝐶𝑅𝐵) → 𝐴𝑅𝐵)) |
28 | 27 | expdimp 456 |
. . . . . 6
⊢ (((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐴𝑅𝐶) → (𝐶𝑅𝐵 → 𝐴𝑅𝐵)) |
29 | 28 | impancom 455 |
. . . . 5
⊢ (((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐶𝑅𝐵) → (𝐴𝑅𝐶 → 𝐴𝑅𝐵)) |
30 | 23, 29 | syldan 594 |
. . . 4
⊢ (((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ ¬ 𝐵𝑅𝐶) → (𝐴𝑅𝐶 → 𝐴𝑅𝐵)) |
31 | 19, 30 | jaod 858 |
. . 3
⊢ (((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ ¬ 𝐵𝑅𝐶) → ((𝐴𝑅𝐵 ∨ 𝐴𝑅𝐶) → 𝐴𝑅𝐵)) |
32 | 18, 31 | impbid2 229 |
. 2
⊢ (((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ ¬ 𝐵𝑅𝐶) → (𝐴𝑅𝐵 ↔ (𝐴𝑅𝐵 ∨ 𝐴𝑅𝐶))) |
33 | 2, 4, 17, 32 | ifbothda 4462 |
1
⊢ ((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝑅if(𝐵𝑅𝐶, 𝐶, 𝐵) ↔ (𝐴𝑅𝐵 ∨ 𝐴𝑅𝐶))) |