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Theorem xpord3lem 8081
Description: Lemma for triple ordering. Calculate the value of the relation. (Contributed by Scott Fenton, 21-Aug-2024.)
Hypothesis
Ref Expression
xpord3.1 𝑈 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ 𝑦 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ ((((1st ‘(1st𝑥))𝑅(1st ‘(1st𝑦)) ∨ (1st ‘(1st𝑥)) = (1st ‘(1st𝑦))) ∧ ((2nd ‘(1st𝑥))𝑆(2nd ‘(1st𝑦)) ∨ (2nd ‘(1st𝑥)) = (2nd ‘(1st𝑦))) ∧ ((2nd𝑥)𝑇(2nd𝑦) ∨ (2nd𝑥) = (2nd𝑦))) ∧ 𝑥𝑦))}
Assertion
Ref Expression
xpord3lem (⟨𝑎, 𝑏, 𝑐𝑈𝑑, 𝑒, 𝑓⟩ ↔ ((𝑎𝐴𝑏𝐵𝑐𝐶) ∧ (𝑑𝐴𝑒𝐵𝑓𝐶) ∧ (((𝑎𝑅𝑑𝑎 = 𝑑) ∧ (𝑏𝑆𝑒𝑏 = 𝑒) ∧ (𝑐𝑇𝑓𝑐 = 𝑓)) ∧ (𝑎𝑑𝑏𝑒𝑐𝑓))))
Distinct variable groups:   𝑥,𝑎,𝑦   𝑥,𝐴,𝑦   𝑥,𝑏,𝑦   𝑥,𝐵,𝑦   𝑥,𝑐,𝑦   𝑥,𝐶,𝑦   𝑥,𝑑,𝑦   𝑥,𝑒,𝑦   𝑥,𝑓,𝑦   𝑥,𝑅,𝑦   𝑥,𝑆,𝑦   𝑥,𝑇,𝑦
Allowed substitution hints:   𝐴(𝑒,𝑓,𝑎,𝑏,𝑐,𝑑)   𝐵(𝑒,𝑓,𝑎,𝑏,𝑐,𝑑)   𝐶(𝑒,𝑓,𝑎,𝑏,𝑐,𝑑)   𝑅(𝑒,𝑓,𝑎,𝑏,𝑐,𝑑)   𝑆(𝑒,𝑓,𝑎,𝑏,𝑐,𝑑)   𝑇(𝑒,𝑓,𝑎,𝑏,𝑐,𝑑)   𝑈(𝑥,𝑦,𝑒,𝑓,𝑎,𝑏,𝑐,𝑑)

Proof of Theorem xpord3lem
StepHypRef Expression
1 otex 5422 . . 3 𝑎, 𝑏, 𝑐⟩ ∈ V
2 otex 5422 . . 3 𝑑, 𝑒, 𝑓⟩ ∈ V
3 eleq1 2825 . . . 4 (𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ → (𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) ↔ ⟨𝑎, 𝑏, 𝑐⟩ ∈ ((𝐴 × 𝐵) × 𝐶)))
4 2fveq3 6847 . . . . . . . . 9 (𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ → (1st ‘(1st𝑥)) = (1st ‘(1st ‘⟨𝑎, 𝑏, 𝑐⟩)))
5 vex 3449 . . . . . . . . . 10 𝑎 ∈ V
6 vex 3449 . . . . . . . . . 10 𝑏 ∈ V
7 vex 3449 . . . . . . . . . 10 𝑐 ∈ V
8 ot1stg 7935 . . . . . . . . . 10 ((𝑎 ∈ V ∧ 𝑏 ∈ V ∧ 𝑐 ∈ V) → (1st ‘(1st ‘⟨𝑎, 𝑏, 𝑐⟩)) = 𝑎)
95, 6, 7, 8mp3an 1461 . . . . . . . . 9 (1st ‘(1st ‘⟨𝑎, 𝑏, 𝑐⟩)) = 𝑎
104, 9eqtrdi 2792 . . . . . . . 8 (𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ → (1st ‘(1st𝑥)) = 𝑎)
1110breq1d 5115 . . . . . . 7 (𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ → ((1st ‘(1st𝑥))𝑅(1st ‘(1st𝑦)) ↔ 𝑎𝑅(1st ‘(1st𝑦))))
1210eqeq1d 2738 . . . . . . 7 (𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ → ((1st ‘(1st𝑥)) = (1st ‘(1st𝑦)) ↔ 𝑎 = (1st ‘(1st𝑦))))
1311, 12orbi12d 917 . . . . . 6 (𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ → (((1st ‘(1st𝑥))𝑅(1st ‘(1st𝑦)) ∨ (1st ‘(1st𝑥)) = (1st ‘(1st𝑦))) ↔ (𝑎𝑅(1st ‘(1st𝑦)) ∨ 𝑎 = (1st ‘(1st𝑦)))))
14 2fveq3 6847 . . . . . . . . 9 (𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ → (2nd ‘(1st𝑥)) = (2nd ‘(1st ‘⟨𝑎, 𝑏, 𝑐⟩)))
15 ot2ndg 7936 . . . . . . . . . 10 ((𝑎 ∈ V ∧ 𝑏 ∈ V ∧ 𝑐 ∈ V) → (2nd ‘(1st ‘⟨𝑎, 𝑏, 𝑐⟩)) = 𝑏)
165, 6, 7, 15mp3an 1461 . . . . . . . . 9 (2nd ‘(1st ‘⟨𝑎, 𝑏, 𝑐⟩)) = 𝑏
1714, 16eqtrdi 2792 . . . . . . . 8 (𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ → (2nd ‘(1st𝑥)) = 𝑏)
1817breq1d 5115 . . . . . . 7 (𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ → ((2nd ‘(1st𝑥))𝑆(2nd ‘(1st𝑦)) ↔ 𝑏𝑆(2nd ‘(1st𝑦))))
1917eqeq1d 2738 . . . . . . 7 (𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ → ((2nd ‘(1st𝑥)) = (2nd ‘(1st𝑦)) ↔ 𝑏 = (2nd ‘(1st𝑦))))
2018, 19orbi12d 917 . . . . . 6 (𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ → (((2nd ‘(1st𝑥))𝑆(2nd ‘(1st𝑦)) ∨ (2nd ‘(1st𝑥)) = (2nd ‘(1st𝑦))) ↔ (𝑏𝑆(2nd ‘(1st𝑦)) ∨ 𝑏 = (2nd ‘(1st𝑦)))))
21 fveq2 6842 . . . . . . . . 9 (𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ → (2nd𝑥) = (2nd ‘⟨𝑎, 𝑏, 𝑐⟩))
22 ot3rdg 7937 . . . . . . . . . 10 (𝑐 ∈ V → (2nd ‘⟨𝑎, 𝑏, 𝑐⟩) = 𝑐)
2322elv 3451 . . . . . . . . 9 (2nd ‘⟨𝑎, 𝑏, 𝑐⟩) = 𝑐
2421, 23eqtrdi 2792 . . . . . . . 8 (𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ → (2nd𝑥) = 𝑐)
2524breq1d 5115 . . . . . . 7 (𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ → ((2nd𝑥)𝑇(2nd𝑦) ↔ 𝑐𝑇(2nd𝑦)))
2624eqeq1d 2738 . . . . . . 7 (𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ → ((2nd𝑥) = (2nd𝑦) ↔ 𝑐 = (2nd𝑦)))
2725, 26orbi12d 917 . . . . . 6 (𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ → (((2nd𝑥)𝑇(2nd𝑦) ∨ (2nd𝑥) = (2nd𝑦)) ↔ (𝑐𝑇(2nd𝑦) ∨ 𝑐 = (2nd𝑦))))
2813, 20, 273anbi123d 1436 . . . . 5 (𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ → ((((1st ‘(1st𝑥))𝑅(1st ‘(1st𝑦)) ∨ (1st ‘(1st𝑥)) = (1st ‘(1st𝑦))) ∧ ((2nd ‘(1st𝑥))𝑆(2nd ‘(1st𝑦)) ∨ (2nd ‘(1st𝑥)) = (2nd ‘(1st𝑦))) ∧ ((2nd𝑥)𝑇(2nd𝑦) ∨ (2nd𝑥) = (2nd𝑦))) ↔ ((𝑎𝑅(1st ‘(1st𝑦)) ∨ 𝑎 = (1st ‘(1st𝑦))) ∧ (𝑏𝑆(2nd ‘(1st𝑦)) ∨ 𝑏 = (2nd ‘(1st𝑦))) ∧ (𝑐𝑇(2nd𝑦) ∨ 𝑐 = (2nd𝑦)))))
29 neeq1 3006 . . . . 5 (𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ → (𝑥𝑦 ↔ ⟨𝑎, 𝑏, 𝑐⟩ ≠ 𝑦))
3028, 29anbi12d 631 . . . 4 (𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ → (((((1st ‘(1st𝑥))𝑅(1st ‘(1st𝑦)) ∨ (1st ‘(1st𝑥)) = (1st ‘(1st𝑦))) ∧ ((2nd ‘(1st𝑥))𝑆(2nd ‘(1st𝑦)) ∨ (2nd ‘(1st𝑥)) = (2nd ‘(1st𝑦))) ∧ ((2nd𝑥)𝑇(2nd𝑦) ∨ (2nd𝑥) = (2nd𝑦))) ∧ 𝑥𝑦) ↔ (((𝑎𝑅(1st ‘(1st𝑦)) ∨ 𝑎 = (1st ‘(1st𝑦))) ∧ (𝑏𝑆(2nd ‘(1st𝑦)) ∨ 𝑏 = (2nd ‘(1st𝑦))) ∧ (𝑐𝑇(2nd𝑦) ∨ 𝑐 = (2nd𝑦))) ∧ ⟨𝑎, 𝑏, 𝑐⟩ ≠ 𝑦)))
313, 303anbi13d 1438 . . 3 (𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ → ((𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ 𝑦 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ ((((1st ‘(1st𝑥))𝑅(1st ‘(1st𝑦)) ∨ (1st ‘(1st𝑥)) = (1st ‘(1st𝑦))) ∧ ((2nd ‘(1st𝑥))𝑆(2nd ‘(1st𝑦)) ∨ (2nd ‘(1st𝑥)) = (2nd ‘(1st𝑦))) ∧ ((2nd𝑥)𝑇(2nd𝑦) ∨ (2nd𝑥) = (2nd𝑦))) ∧ 𝑥𝑦)) ↔ (⟨𝑎, 𝑏, 𝑐⟩ ∈ ((𝐴 × 𝐵) × 𝐶) ∧ 𝑦 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ (((𝑎𝑅(1st ‘(1st𝑦)) ∨ 𝑎 = (1st ‘(1st𝑦))) ∧ (𝑏𝑆(2nd ‘(1st𝑦)) ∨ 𝑏 = (2nd ‘(1st𝑦))) ∧ (𝑐𝑇(2nd𝑦) ∨ 𝑐 = (2nd𝑦))) ∧ ⟨𝑎, 𝑏, 𝑐⟩ ≠ 𝑦))))
32 eleq1 2825 . . . 4 (𝑦 = ⟨𝑑, 𝑒, 𝑓⟩ → (𝑦 ∈ ((𝐴 × 𝐵) × 𝐶) ↔ ⟨𝑑, 𝑒, 𝑓⟩ ∈ ((𝐴 × 𝐵) × 𝐶)))
33 2fveq3 6847 . . . . . . . . 9 (𝑦 = ⟨𝑑, 𝑒, 𝑓⟩ → (1st ‘(1st𝑦)) = (1st ‘(1st ‘⟨𝑑, 𝑒, 𝑓⟩)))
34 vex 3449 . . . . . . . . . 10 𝑑 ∈ V
35 vex 3449 . . . . . . . . . 10 𝑒 ∈ V
36 vex 3449 . . . . . . . . . 10 𝑓 ∈ V
37 ot1stg 7935 . . . . . . . . . 10 ((𝑑 ∈ V ∧ 𝑒 ∈ V ∧ 𝑓 ∈ V) → (1st ‘(1st ‘⟨𝑑, 𝑒, 𝑓⟩)) = 𝑑)
3834, 35, 36, 37mp3an 1461 . . . . . . . . 9 (1st ‘(1st ‘⟨𝑑, 𝑒, 𝑓⟩)) = 𝑑
3933, 38eqtrdi 2792 . . . . . . . 8 (𝑦 = ⟨𝑑, 𝑒, 𝑓⟩ → (1st ‘(1st𝑦)) = 𝑑)
4039breq2d 5117 . . . . . . 7 (𝑦 = ⟨𝑑, 𝑒, 𝑓⟩ → (𝑎𝑅(1st ‘(1st𝑦)) ↔ 𝑎𝑅𝑑))
4139eqeq2d 2747 . . . . . . 7 (𝑦 = ⟨𝑑, 𝑒, 𝑓⟩ → (𝑎 = (1st ‘(1st𝑦)) ↔ 𝑎 = 𝑑))
4240, 41orbi12d 917 . . . . . 6 (𝑦 = ⟨𝑑, 𝑒, 𝑓⟩ → ((𝑎𝑅(1st ‘(1st𝑦)) ∨ 𝑎 = (1st ‘(1st𝑦))) ↔ (𝑎𝑅𝑑𝑎 = 𝑑)))
43 2fveq3 6847 . . . . . . . . 9 (𝑦 = ⟨𝑑, 𝑒, 𝑓⟩ → (2nd ‘(1st𝑦)) = (2nd ‘(1st ‘⟨𝑑, 𝑒, 𝑓⟩)))
44 ot2ndg 7936 . . . . . . . . . 10 ((𝑑 ∈ V ∧ 𝑒 ∈ V ∧ 𝑓 ∈ V) → (2nd ‘(1st ‘⟨𝑑, 𝑒, 𝑓⟩)) = 𝑒)
4534, 35, 36, 44mp3an 1461 . . . . . . . . 9 (2nd ‘(1st ‘⟨𝑑, 𝑒, 𝑓⟩)) = 𝑒
4643, 45eqtrdi 2792 . . . . . . . 8 (𝑦 = ⟨𝑑, 𝑒, 𝑓⟩ → (2nd ‘(1st𝑦)) = 𝑒)
4746breq2d 5117 . . . . . . 7 (𝑦 = ⟨𝑑, 𝑒, 𝑓⟩ → (𝑏𝑆(2nd ‘(1st𝑦)) ↔ 𝑏𝑆𝑒))
4846eqeq2d 2747 . . . . . . 7 (𝑦 = ⟨𝑑, 𝑒, 𝑓⟩ → (𝑏 = (2nd ‘(1st𝑦)) ↔ 𝑏 = 𝑒))
4947, 48orbi12d 917 . . . . . 6 (𝑦 = ⟨𝑑, 𝑒, 𝑓⟩ → ((𝑏𝑆(2nd ‘(1st𝑦)) ∨ 𝑏 = (2nd ‘(1st𝑦))) ↔ (𝑏𝑆𝑒𝑏 = 𝑒)))
50 fveq2 6842 . . . . . . . . 9 (𝑦 = ⟨𝑑, 𝑒, 𝑓⟩ → (2nd𝑦) = (2nd ‘⟨𝑑, 𝑒, 𝑓⟩))
51 ot3rdg 7937 . . . . . . . . . 10 (𝑓 ∈ V → (2nd ‘⟨𝑑, 𝑒, 𝑓⟩) = 𝑓)
5251elv 3451 . . . . . . . . 9 (2nd ‘⟨𝑑, 𝑒, 𝑓⟩) = 𝑓
5350, 52eqtrdi 2792 . . . . . . . 8 (𝑦 = ⟨𝑑, 𝑒, 𝑓⟩ → (2nd𝑦) = 𝑓)
5453breq2d 5117 . . . . . . 7 (𝑦 = ⟨𝑑, 𝑒, 𝑓⟩ → (𝑐𝑇(2nd𝑦) ↔ 𝑐𝑇𝑓))
5553eqeq2d 2747 . . . . . . 7 (𝑦 = ⟨𝑑, 𝑒, 𝑓⟩ → (𝑐 = (2nd𝑦) ↔ 𝑐 = 𝑓))
5654, 55orbi12d 917 . . . . . 6 (𝑦 = ⟨𝑑, 𝑒, 𝑓⟩ → ((𝑐𝑇(2nd𝑦) ∨ 𝑐 = (2nd𝑦)) ↔ (𝑐𝑇𝑓𝑐 = 𝑓)))
5742, 49, 563anbi123d 1436 . . . . 5 (𝑦 = ⟨𝑑, 𝑒, 𝑓⟩ → (((𝑎𝑅(1st ‘(1st𝑦)) ∨ 𝑎 = (1st ‘(1st𝑦))) ∧ (𝑏𝑆(2nd ‘(1st𝑦)) ∨ 𝑏 = (2nd ‘(1st𝑦))) ∧ (𝑐𝑇(2nd𝑦) ∨ 𝑐 = (2nd𝑦))) ↔ ((𝑎𝑅𝑑𝑎 = 𝑑) ∧ (𝑏𝑆𝑒𝑏 = 𝑒) ∧ (𝑐𝑇𝑓𝑐 = 𝑓))))
58 neeq2 3007 . . . . 5 (𝑦 = ⟨𝑑, 𝑒, 𝑓⟩ → (⟨𝑎, 𝑏, 𝑐⟩ ≠ 𝑦 ↔ ⟨𝑎, 𝑏, 𝑐⟩ ≠ ⟨𝑑, 𝑒, 𝑓⟩))
5957, 58anbi12d 631 . . . 4 (𝑦 = ⟨𝑑, 𝑒, 𝑓⟩ → ((((𝑎𝑅(1st ‘(1st𝑦)) ∨ 𝑎 = (1st ‘(1st𝑦))) ∧ (𝑏𝑆(2nd ‘(1st𝑦)) ∨ 𝑏 = (2nd ‘(1st𝑦))) ∧ (𝑐𝑇(2nd𝑦) ∨ 𝑐 = (2nd𝑦))) ∧ ⟨𝑎, 𝑏, 𝑐⟩ ≠ 𝑦) ↔ (((𝑎𝑅𝑑𝑎 = 𝑑) ∧ (𝑏𝑆𝑒𝑏 = 𝑒) ∧ (𝑐𝑇𝑓𝑐 = 𝑓)) ∧ ⟨𝑎, 𝑏, 𝑐⟩ ≠ ⟨𝑑, 𝑒, 𝑓⟩)))
6032, 593anbi23d 1439 . . 3 (𝑦 = ⟨𝑑, 𝑒, 𝑓⟩ → ((⟨𝑎, 𝑏, 𝑐⟩ ∈ ((𝐴 × 𝐵) × 𝐶) ∧ 𝑦 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ (((𝑎𝑅(1st ‘(1st𝑦)) ∨ 𝑎 = (1st ‘(1st𝑦))) ∧ (𝑏𝑆(2nd ‘(1st𝑦)) ∨ 𝑏 = (2nd ‘(1st𝑦))) ∧ (𝑐𝑇(2nd𝑦) ∨ 𝑐 = (2nd𝑦))) ∧ ⟨𝑎, 𝑏, 𝑐⟩ ≠ 𝑦)) ↔ (⟨𝑎, 𝑏, 𝑐⟩ ∈ ((𝐴 × 𝐵) × 𝐶) ∧ ⟨𝑑, 𝑒, 𝑓⟩ ∈ ((𝐴 × 𝐵) × 𝐶) ∧ (((𝑎𝑅𝑑𝑎 = 𝑑) ∧ (𝑏𝑆𝑒𝑏 = 𝑒) ∧ (𝑐𝑇𝑓𝑐 = 𝑓)) ∧ ⟨𝑎, 𝑏, 𝑐⟩ ≠ ⟨𝑑, 𝑒, 𝑓⟩))))
61 xpord3.1 . . 3 𝑈 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ 𝑦 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ ((((1st ‘(1st𝑥))𝑅(1st ‘(1st𝑦)) ∨ (1st ‘(1st𝑥)) = (1st ‘(1st𝑦))) ∧ ((2nd ‘(1st𝑥))𝑆(2nd ‘(1st𝑦)) ∨ (2nd ‘(1st𝑥)) = (2nd ‘(1st𝑦))) ∧ ((2nd𝑥)𝑇(2nd𝑦) ∨ (2nd𝑥) = (2nd𝑦))) ∧ 𝑥𝑦))}
621, 2, 31, 60, 61brab 5500 . 2 (⟨𝑎, 𝑏, 𝑐𝑈𝑑, 𝑒, 𝑓⟩ ↔ (⟨𝑎, 𝑏, 𝑐⟩ ∈ ((𝐴 × 𝐵) × 𝐶) ∧ ⟨𝑑, 𝑒, 𝑓⟩ ∈ ((𝐴 × 𝐵) × 𝐶) ∧ (((𝑎𝑅𝑑𝑎 = 𝑑) ∧ (𝑏𝑆𝑒𝑏 = 𝑒) ∧ (𝑐𝑇𝑓𝑐 = 𝑓)) ∧ ⟨𝑎, 𝑏, 𝑐⟩ ≠ ⟨𝑑, 𝑒, 𝑓⟩)))
63 otelxp 5676 . . 3 (⟨𝑎, 𝑏, 𝑐⟩ ∈ ((𝐴 × 𝐵) × 𝐶) ↔ (𝑎𝐴𝑏𝐵𝑐𝐶))
64 otelxp 5676 . . 3 (⟨𝑑, 𝑒, 𝑓⟩ ∈ ((𝐴 × 𝐵) × 𝐶) ↔ (𝑑𝐴𝑒𝐵𝑓𝐶))
655, 6, 7otthne 5443 . . . 4 (⟨𝑎, 𝑏, 𝑐⟩ ≠ ⟨𝑑, 𝑒, 𝑓⟩ ↔ (𝑎𝑑𝑏𝑒𝑐𝑓))
6665anbi2i 623 . . 3 ((((𝑎𝑅𝑑𝑎 = 𝑑) ∧ (𝑏𝑆𝑒𝑏 = 𝑒) ∧ (𝑐𝑇𝑓𝑐 = 𝑓)) ∧ ⟨𝑎, 𝑏, 𝑐⟩ ≠ ⟨𝑑, 𝑒, 𝑓⟩) ↔ (((𝑎𝑅𝑑𝑎 = 𝑑) ∧ (𝑏𝑆𝑒𝑏 = 𝑒) ∧ (𝑐𝑇𝑓𝑐 = 𝑓)) ∧ (𝑎𝑑𝑏𝑒𝑐𝑓)))
6763, 64, 663anbi123i 1155 . 2 ((⟨𝑎, 𝑏, 𝑐⟩ ∈ ((𝐴 × 𝐵) × 𝐶) ∧ ⟨𝑑, 𝑒, 𝑓⟩ ∈ ((𝐴 × 𝐵) × 𝐶) ∧ (((𝑎𝑅𝑑𝑎 = 𝑑) ∧ (𝑏𝑆𝑒𝑏 = 𝑒) ∧ (𝑐𝑇𝑓𝑐 = 𝑓)) ∧ ⟨𝑎, 𝑏, 𝑐⟩ ≠ ⟨𝑑, 𝑒, 𝑓⟩)) ↔ ((𝑎𝐴𝑏𝐵𝑐𝐶) ∧ (𝑑𝐴𝑒𝐵𝑓𝐶) ∧ (((𝑎𝑅𝑑𝑎 = 𝑑) ∧ (𝑏𝑆𝑒𝑏 = 𝑒) ∧ (𝑐𝑇𝑓𝑐 = 𝑓)) ∧ (𝑎𝑑𝑏𝑒𝑐𝑓))))
6862, 67bitri 274 1 (⟨𝑎, 𝑏, 𝑐𝑈𝑑, 𝑒, 𝑓⟩ ↔ ((𝑎𝐴𝑏𝐵𝑐𝐶) ∧ (𝑑𝐴𝑒𝐵𝑓𝐶) ∧ (((𝑎𝑅𝑑𝑎 = 𝑑) ∧ (𝑏𝑆𝑒𝑏 = 𝑒) ∧ (𝑐𝑇𝑓𝑐 = 𝑓)) ∧ (𝑎𝑑𝑏𝑒𝑐𝑓))))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396  wo 845  w3o 1086  w3a 1087   = wceq 1541  wcel 2106  wne 2943  Vcvv 3445  cotp 4594   class class class wbr 5105  {copab 5167   × cxp 5631  cfv 6496  1st c1st 7919  2nd c2nd 7920
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-ot 4595  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-iota 6448  df-fun 6498  df-fv 6504  df-1st 7921  df-2nd 7922
This theorem is referenced by:  poxp3  8082  frxp3  8083  xpord3pred  8084
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