Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  prproropf1olem4 Structured version   Visualization version   GIF version

Theorem prproropf1olem4 44066
 Description: Lemma 4 for prproropf1o 44067. (Contributed by AV, 14-Mar-2023.)
Hypotheses
Ref Expression
prproropf1o.o 𝑂 = (𝑅 ∩ (𝑉 × 𝑉))
prproropf1o.p 𝑃 = {𝑝 ∈ 𝒫 𝑉 ∣ (♯‘𝑝) = 2}
prproropf1o.f 𝐹 = (𝑝𝑃 ↦ ⟨inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)⟩)
Assertion
Ref Expression
prproropf1olem4 ((𝑅 Or 𝑉𝑊𝑃𝑍𝑃) → ((𝐹𝑍) = (𝐹𝑊) → 𝑍 = 𝑊))
Distinct variable groups:   𝑉,𝑝   𝑊,𝑝   𝑂,𝑝   𝑃,𝑝   𝑅,𝑝   𝑍,𝑝
Allowed substitution hint:   𝐹(𝑝)

Proof of Theorem prproropf1olem4
Dummy variables 𝑎 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prproropf1o.f . . . 4 𝐹 = (𝑝𝑃 ↦ ⟨inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)⟩)
2 infeq1 8927 . . . . 5 (𝑝 = 𝑍 → inf(𝑝, 𝑉, 𝑅) = inf(𝑍, 𝑉, 𝑅))
3 supeq1 8896 . . . . 5 (𝑝 = 𝑍 → sup(𝑝, 𝑉, 𝑅) = sup(𝑍, 𝑉, 𝑅))
42, 3opeq12d 4774 . . . 4 (𝑝 = 𝑍 → ⟨inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)⟩ = ⟨inf(𝑍, 𝑉, 𝑅), sup(𝑍, 𝑉, 𝑅)⟩)
5 simp3 1135 . . . 4 ((𝑅 Or 𝑉𝑊𝑃𝑍𝑃) → 𝑍𝑃)
6 opex 5322 . . . . 5 ⟨inf(𝑍, 𝑉, 𝑅), sup(𝑍, 𝑉, 𝑅)⟩ ∈ V
76a1i 11 . . . 4 ((𝑅 Or 𝑉𝑊𝑃𝑍𝑃) → ⟨inf(𝑍, 𝑉, 𝑅), sup(𝑍, 𝑉, 𝑅)⟩ ∈ V)
81, 4, 5, 7fvmptd3 6769 . . 3 ((𝑅 Or 𝑉𝑊𝑃𝑍𝑃) → (𝐹𝑍) = ⟨inf(𝑍, 𝑉, 𝑅), sup(𝑍, 𝑉, 𝑅)⟩)
9 infeq1 8927 . . . . 5 (𝑝 = 𝑊 → inf(𝑝, 𝑉, 𝑅) = inf(𝑊, 𝑉, 𝑅))
10 supeq1 8896 . . . . 5 (𝑝 = 𝑊 → sup(𝑝, 𝑉, 𝑅) = sup(𝑊, 𝑉, 𝑅))
119, 10opeq12d 4774 . . . 4 (𝑝 = 𝑊 → ⟨inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)⟩ = ⟨inf(𝑊, 𝑉, 𝑅), sup(𝑊, 𝑉, 𝑅)⟩)
12 simp2 1134 . . . 4 ((𝑅 Or 𝑉𝑊𝑃𝑍𝑃) → 𝑊𝑃)
13 opex 5322 . . . . 5 ⟨inf(𝑊, 𝑉, 𝑅), sup(𝑊, 𝑉, 𝑅)⟩ ∈ V
1413a1i 11 . . . 4 ((𝑅 Or 𝑉𝑊𝑃𝑍𝑃) → ⟨inf(𝑊, 𝑉, 𝑅), sup(𝑊, 𝑉, 𝑅)⟩ ∈ V)
151, 11, 12, 14fvmptd3 6769 . . 3 ((𝑅 Or 𝑉𝑊𝑃𝑍𝑃) → (𝐹𝑊) = ⟨inf(𝑊, 𝑉, 𝑅), sup(𝑊, 𝑉, 𝑅)⟩)
168, 15eqeq12d 2814 . 2 ((𝑅 Or 𝑉𝑊𝑃𝑍𝑃) → ((𝐹𝑍) = (𝐹𝑊) ↔ ⟨inf(𝑍, 𝑉, 𝑅), sup(𝑍, 𝑉, 𝑅)⟩ = ⟨inf(𝑊, 𝑉, 𝑅), sup(𝑊, 𝑉, 𝑅)⟩))
17 prproropf1o.p . . . . 5 𝑃 = {𝑝 ∈ 𝒫 𝑉 ∣ (♯‘𝑝) = 2}
1817prpair 44061 . . . 4 (𝑍𝑃 ↔ ∃𝑐𝑉𝑑𝑉 (𝑍 = {𝑐, 𝑑} ∧ 𝑐𝑑))
1917prpair 44061 . . . . 5 (𝑊𝑃 ↔ ∃𝑎𝑉𝑏𝑉 (𝑊 = {𝑎, 𝑏} ∧ 𝑎𝑏))
20 id 22 . . . . . . . . . . . . . . . 16 (𝑅 Or 𝑉𝑅 Or 𝑉)
2120infexd 8934 . . . . . . . . . . . . . . 15 (𝑅 Or 𝑉 → inf({𝑐, 𝑑}, 𝑉, 𝑅) ∈ V)
2220supexd 8904 . . . . . . . . . . . . . . 15 (𝑅 Or 𝑉 → sup({𝑐, 𝑑}, 𝑉, 𝑅) ∈ V)
2321, 22jca 515 . . . . . . . . . . . . . 14 (𝑅 Or 𝑉 → (inf({𝑐, 𝑑}, 𝑉, 𝑅) ∈ V ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) ∈ V))
2423ad4antr 731 . . . . . . . . . . . . 13 (((((𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑊 = {𝑎, 𝑏} ∧ 𝑎𝑏)) ∧ (𝑐𝑉𝑑𝑉)) ∧ (𝑍 = {𝑐, 𝑑} ∧ 𝑐𝑑)) → (inf({𝑐, 𝑑}, 𝑉, 𝑅) ∈ V ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) ∈ V))
25 opthg 5335 . . . . . . . . . . . . 13 ((inf({𝑐, 𝑑}, 𝑉, 𝑅) ∈ V ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) ∈ V) → (⟨inf({𝑐, 𝑑}, 𝑉, 𝑅), sup({𝑐, 𝑑}, 𝑉, 𝑅)⟩ = ⟨inf({𝑎, 𝑏}, 𝑉, 𝑅), sup({𝑎, 𝑏}, 𝑉, 𝑅)⟩ ↔ (inf({𝑐, 𝑑}, 𝑉, 𝑅) = inf({𝑎, 𝑏}, 𝑉, 𝑅) ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = sup({𝑎, 𝑏}, 𝑉, 𝑅))))
2624, 25syl 17 . . . . . . . . . . . 12 (((((𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑊 = {𝑎, 𝑏} ∧ 𝑎𝑏)) ∧ (𝑐𝑉𝑑𝑉)) ∧ (𝑍 = {𝑐, 𝑑} ∧ 𝑐𝑑)) → (⟨inf({𝑐, 𝑑}, 𝑉, 𝑅), sup({𝑐, 𝑑}, 𝑉, 𝑅)⟩ = ⟨inf({𝑎, 𝑏}, 𝑉, 𝑅), sup({𝑎, 𝑏}, 𝑉, 𝑅)⟩ ↔ (inf({𝑐, 𝑑}, 𝑉, 𝑅) = inf({𝑎, 𝑏}, 𝑉, 𝑅) ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = sup({𝑎, 𝑏}, 𝑉, 𝑅))))
27 solin 5463 . . . . . . . . . . . . . . . . . . 19 ((𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉)) → (𝑎𝑅𝑏𝑎 = 𝑏𝑏𝑅𝑎))
28 infpr 8954 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑅 Or 𝑉𝑎𝑉𝑏𝑉) → inf({𝑎, 𝑏}, 𝑉, 𝑅) = if(𝑎𝑅𝑏, 𝑎, 𝑏))
29283expb 1117 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉)) → inf({𝑎, 𝑏}, 𝑉, 𝑅) = if(𝑎𝑅𝑏, 𝑎, 𝑏))
30 iftrue 4431 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑎𝑅𝑏 → if(𝑎𝑅𝑏, 𝑎, 𝑏) = 𝑎)
3129, 30sylan9eqr 2855 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑎𝑅𝑏 ∧ (𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉))) → inf({𝑎, 𝑏}, 𝑉, 𝑅) = 𝑎)
3231eqeq2d 2809 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑎𝑅𝑏 ∧ (𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉))) → (inf({𝑐, 𝑑}, 𝑉, 𝑅) = inf({𝑎, 𝑏}, 𝑉, 𝑅) ↔ inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎))
33 suppr 8922 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑅 Or 𝑉𝑎𝑉𝑏𝑉) → sup({𝑎, 𝑏}, 𝑉, 𝑅) = if(𝑏𝑅𝑎, 𝑎, 𝑏))
34333expb 1117 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉)) → sup({𝑎, 𝑏}, 𝑉, 𝑅) = if(𝑏𝑅𝑎, 𝑎, 𝑏))
3534adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑎𝑅𝑏 ∧ (𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉))) → sup({𝑎, 𝑏}, 𝑉, 𝑅) = if(𝑏𝑅𝑎, 𝑎, 𝑏))
36 sotric 5466 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉)) → (𝑎𝑅𝑏 ↔ ¬ (𝑎 = 𝑏𝑏𝑅𝑎)))
37 ioran 981 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (¬ (𝑎 = 𝑏𝑏𝑅𝑎) ↔ (¬ 𝑎 = 𝑏 ∧ ¬ 𝑏𝑅𝑎))
38 iffalse 4434 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 𝑏𝑅𝑎 → if(𝑏𝑅𝑎, 𝑎, 𝑏) = 𝑏)
3937, 38simplbiim 508 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (¬ (𝑎 = 𝑏𝑏𝑅𝑎) → if(𝑏𝑅𝑎, 𝑎, 𝑏) = 𝑏)
4036, 39syl6bi 256 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉)) → (𝑎𝑅𝑏 → if(𝑏𝑅𝑎, 𝑎, 𝑏) = 𝑏))
4140impcom 411 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑎𝑅𝑏 ∧ (𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉))) → if(𝑏𝑅𝑎, 𝑎, 𝑏) = 𝑏)
4235, 41eqtrd 2833 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑎𝑅𝑏 ∧ (𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉))) → sup({𝑎, 𝑏}, 𝑉, 𝑅) = 𝑏)
4342eqeq2d 2809 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑎𝑅𝑏 ∧ (𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉))) → (sup({𝑐, 𝑑}, 𝑉, 𝑅) = sup({𝑎, 𝑏}, 𝑉, 𝑅) ↔ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏))
4432, 43anbi12d 633 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑎𝑅𝑏 ∧ (𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉))) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = inf({𝑎, 𝑏}, 𝑉, 𝑅) ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = sup({𝑎, 𝑏}, 𝑉, 𝑅)) ↔ (inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏)))
4544adantr 484 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑎𝑅𝑏 ∧ (𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉))) ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑)) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = inf({𝑎, 𝑏}, 𝑉, 𝑅) ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = sup({𝑎, 𝑏}, 𝑉, 𝑅)) ↔ (inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏)))
46 solin 5463 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑅 Or 𝑉 ∧ (𝑐𝑉𝑑𝑉)) → (𝑐𝑅𝑑𝑐 = 𝑑𝑑𝑅𝑐))
4746adantrr 716 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑)) → (𝑐𝑅𝑑𝑐 = 𝑑𝑑𝑅𝑐))
48 simpl 486 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑)) → 𝑅 Or 𝑉)
49 simprll 778 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑)) → 𝑐𝑉)
50 simprlr 779 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑)) → 𝑑𝑉)
51 infpr 8954 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑅 Or 𝑉𝑐𝑉𝑑𝑉) → inf({𝑐, 𝑑}, 𝑉, 𝑅) = if(𝑐𝑅𝑑, 𝑐, 𝑑))
5248, 49, 50, 51syl3anc 1368 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑)) → inf({𝑐, 𝑑}, 𝑉, 𝑅) = if(𝑐𝑅𝑑, 𝑐, 𝑑))
53 iftrue 4431 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑐𝑅𝑑 → if(𝑐𝑅𝑑, 𝑐, 𝑑) = 𝑐)
5452, 53sylan9eqr 2855 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑐𝑅𝑑 ∧ (𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑))) → inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑐)
5554eqeq1d 2800 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑐𝑅𝑑 ∧ (𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑))) → (inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎𝑐 = 𝑎))
56 suppr 8922 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑅 Or 𝑉𝑐𝑉𝑑𝑉) → sup({𝑐, 𝑑}, 𝑉, 𝑅) = if(𝑑𝑅𝑐, 𝑐, 𝑑))
5748, 49, 50, 56syl3anc 1368 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑)) → sup({𝑐, 𝑑}, 𝑉, 𝑅) = if(𝑑𝑅𝑐, 𝑐, 𝑑))
5857adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑐𝑅𝑑 ∧ (𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑))) → sup({𝑐, 𝑑}, 𝑉, 𝑅) = if(𝑑𝑅𝑐, 𝑐, 𝑑))
59 sotric 5466 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑅 Or 𝑉 ∧ (𝑐𝑉𝑑𝑉)) → (𝑐𝑅𝑑 ↔ ¬ (𝑐 = 𝑑𝑑𝑅𝑐)))
6059adantrr 716 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑)) → (𝑐𝑅𝑑 ↔ ¬ (𝑐 = 𝑑𝑑𝑅𝑐)))
61 ioran 981 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (¬ (𝑐 = 𝑑𝑑𝑅𝑐) ↔ (¬ 𝑐 = 𝑑 ∧ ¬ 𝑑𝑅𝑐))
62 iffalse 4434 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 𝑑𝑅𝑐 → if(𝑑𝑅𝑐, 𝑐, 𝑑) = 𝑑)
6361, 62simplbiim 508 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (¬ (𝑐 = 𝑑𝑑𝑅𝑐) → if(𝑑𝑅𝑐, 𝑐, 𝑑) = 𝑑)
6460, 63syl6bi 256 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑)) → (𝑐𝑅𝑑 → if(𝑑𝑅𝑐, 𝑐, 𝑑) = 𝑑))
6564impcom 411 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑐𝑅𝑑 ∧ (𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑))) → if(𝑑𝑅𝑐, 𝑐, 𝑑) = 𝑑)
6658, 65eqtrd 2833 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑐𝑅𝑑 ∧ (𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑))) → sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑑)
6766eqeq1d 2800 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑐𝑅𝑑 ∧ (𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑))) → (sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏𝑑 = 𝑏))
6855, 67anbi12d 633 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑐𝑅𝑑 ∧ (𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑))) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏) ↔ (𝑐 = 𝑎𝑑 = 𝑏)))
69 orc 864 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑐 = 𝑎𝑑 = 𝑏) → ((𝑐 = 𝑎𝑑 = 𝑏) ∨ (𝑐 = 𝑏𝑑 = 𝑎)))
7068, 69syl6bi 256 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑐𝑅𝑑 ∧ (𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑))) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏) → ((𝑐 = 𝑎𝑑 = 𝑏) ∨ (𝑐 = 𝑏𝑑 = 𝑎))))
7170ex 416 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑐𝑅𝑑 → ((𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑)) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏) → ((𝑐 = 𝑎𝑑 = 𝑏) ∨ (𝑐 = 𝑏𝑑 = 𝑎)))))
72 eqneqall 2998 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑐 = 𝑑 → (𝑐𝑑 → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏) → ((𝑐 = 𝑎𝑑 = 𝑏) ∨ (𝑐 = 𝑏𝑑 = 𝑎)))))
7372adantld 494 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑐 = 𝑑 → (((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏) → ((𝑐 = 𝑎𝑑 = 𝑏) ∨ (𝑐 = 𝑏𝑑 = 𝑎)))))
7473adantld 494 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑐 = 𝑑 → ((𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑)) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏) → ((𝑐 = 𝑎𝑑 = 𝑏) ∨ (𝑐 = 𝑏𝑑 = 𝑎)))))
7552adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑑𝑅𝑐 ∧ (𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑))) → inf({𝑐, 𝑑}, 𝑉, 𝑅) = if(𝑐𝑅𝑑, 𝑐, 𝑑))
7675eqeq1d 2800 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑑𝑅𝑐 ∧ (𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑))) → (inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎 ↔ if(𝑐𝑅𝑑, 𝑐, 𝑑) = 𝑎))
77 iftrue 4431 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑑𝑅𝑐 → if(𝑑𝑅𝑐, 𝑐, 𝑑) = 𝑐)
7857, 77sylan9eqr 2855 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑑𝑅𝑐 ∧ (𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑))) → sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑐)
7978eqeq1d 2800 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑑𝑅𝑐 ∧ (𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑))) → (sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏𝑐 = 𝑏))
8076, 79anbi12d 633 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑑𝑅𝑐 ∧ (𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑))) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏) ↔ (if(𝑐𝑅𝑑, 𝑐, 𝑑) = 𝑎𝑐 = 𝑏)))
81 simpl 486 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑) → (𝑐𝑉𝑑𝑉))
8281ancomd 465 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑) → (𝑑𝑉𝑐𝑉))
83 sotric 5466 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑅 Or 𝑉 ∧ (𝑑𝑉𝑐𝑉)) → (𝑑𝑅𝑐 ↔ ¬ (𝑑 = 𝑐𝑐𝑅𝑑)))
8482, 83sylan2 595 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑)) → (𝑑𝑅𝑐 ↔ ¬ (𝑑 = 𝑐𝑐𝑅𝑑)))
85 ioran 981 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (¬ (𝑑 = 𝑐𝑐𝑅𝑑) ↔ (¬ 𝑑 = 𝑐 ∧ ¬ 𝑐𝑅𝑑))
86 iffalse 4434 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 𝑐𝑅𝑑 → if(𝑐𝑅𝑑, 𝑐, 𝑑) = 𝑑)
8785, 86simplbiim 508 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (¬ (𝑑 = 𝑐𝑐𝑅𝑑) → if(𝑐𝑅𝑑, 𝑐, 𝑑) = 𝑑)
8887eqeq1d 2800 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (¬ (𝑑 = 𝑐𝑐𝑅𝑑) → (if(𝑐𝑅𝑑, 𝑐, 𝑑) = 𝑎𝑑 = 𝑎))
8984, 88syl6bi 256 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑)) → (𝑑𝑅𝑐 → (if(𝑐𝑅𝑑, 𝑐, 𝑑) = 𝑎𝑑 = 𝑎)))
9089impcom 411 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑑𝑅𝑐 ∧ (𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑))) → (if(𝑐𝑅𝑑, 𝑐, 𝑑) = 𝑎𝑑 = 𝑎))
9190anbi1d 632 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑑𝑅𝑐 ∧ (𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑))) → ((if(𝑐𝑅𝑑, 𝑐, 𝑑) = 𝑎𝑐 = 𝑏) ↔ (𝑑 = 𝑎𝑐 = 𝑏)))
92 olc 865 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑐 = 𝑏𝑑 = 𝑎) → ((𝑐 = 𝑎𝑑 = 𝑏) ∨ (𝑐 = 𝑏𝑑 = 𝑎)))
9392ancoms 462 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑑 = 𝑎𝑐 = 𝑏) → ((𝑐 = 𝑎𝑑 = 𝑏) ∨ (𝑐 = 𝑏𝑑 = 𝑎)))
9491, 93syl6bi 256 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑑𝑅𝑐 ∧ (𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑))) → ((if(𝑐𝑅𝑑, 𝑐, 𝑑) = 𝑎𝑐 = 𝑏) → ((𝑐 = 𝑎𝑑 = 𝑏) ∨ (𝑐 = 𝑏𝑑 = 𝑎))))
9580, 94sylbid 243 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑑𝑅𝑐 ∧ (𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑))) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏) → ((𝑐 = 𝑎𝑑 = 𝑏) ∨ (𝑐 = 𝑏𝑑 = 𝑎))))
9695ex 416 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑑𝑅𝑐 → ((𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑)) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏) → ((𝑐 = 𝑎𝑑 = 𝑏) ∨ (𝑐 = 𝑏𝑑 = 𝑎)))))
9771, 74, 963jaoi 1424 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑐𝑅𝑑𝑐 = 𝑑𝑑𝑅𝑐) → ((𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑)) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏) → ((𝑐 = 𝑎𝑑 = 𝑏) ∨ (𝑐 = 𝑏𝑑 = 𝑎)))))
9847, 97mpcom 38 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑)) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏) → ((𝑐 = 𝑎𝑑 = 𝑏) ∨ (𝑐 = 𝑏𝑑 = 𝑎))))
9998ex 416 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑅 Or 𝑉 → (((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏) → ((𝑐 = 𝑎𝑑 = 𝑏) ∨ (𝑐 = 𝑏𝑑 = 𝑎)))))
10099ad2antrl 727 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑎𝑅𝑏 ∧ (𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉))) → (((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏) → ((𝑐 = 𝑎𝑑 = 𝑏) ∨ (𝑐 = 𝑏𝑑 = 𝑎)))))
101100imp 410 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑎𝑅𝑏 ∧ (𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉))) ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑)) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏) → ((𝑐 = 𝑎𝑑 = 𝑏) ∨ (𝑐 = 𝑏𝑑 = 𝑎))))
10245, 101sylbid 243 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑎𝑅𝑏 ∧ (𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉))) ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑)) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = inf({𝑎, 𝑏}, 𝑉, 𝑅) ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = sup({𝑎, 𝑏}, 𝑉, 𝑅)) → ((𝑐 = 𝑎𝑑 = 𝑏) ∨ (𝑐 = 𝑏𝑑 = 𝑎))))
103102ex 416 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑎𝑅𝑏 ∧ (𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉))) → (((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = inf({𝑎, 𝑏}, 𝑉, 𝑅) ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = sup({𝑎, 𝑏}, 𝑉, 𝑅)) → ((𝑐 = 𝑎𝑑 = 𝑏) ∨ (𝑐 = 𝑏𝑑 = 𝑎)))))
104103a1d 25 . . . . . . . . . . . . . . . . . . . . 21 ((𝑎𝑅𝑏 ∧ (𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉))) → (𝑎𝑏 → (((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = inf({𝑎, 𝑏}, 𝑉, 𝑅) ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = sup({𝑎, 𝑏}, 𝑉, 𝑅)) → ((𝑐 = 𝑎𝑑 = 𝑏) ∨ (𝑐 = 𝑏𝑑 = 𝑎))))))
105104ex 416 . . . . . . . . . . . . . . . . . . . 20 (𝑎𝑅𝑏 → ((𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉)) → (𝑎𝑏 → (((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = inf({𝑎, 𝑏}, 𝑉, 𝑅) ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = sup({𝑎, 𝑏}, 𝑉, 𝑅)) → ((𝑐 = 𝑎𝑑 = 𝑏) ∨ (𝑐 = 𝑏𝑑 = 𝑎)))))))
106 eqneqall 2998 . . . . . . . . . . . . . . . . . . . . 21 (𝑎 = 𝑏 → (𝑎𝑏 → (((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = inf({𝑎, 𝑏}, 𝑉, 𝑅) ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = sup({𝑎, 𝑏}, 𝑉, 𝑅)) → ((𝑐 = 𝑎𝑑 = 𝑏) ∨ (𝑐 = 𝑏𝑑 = 𝑎))))))
107106a1d 25 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = 𝑏 → ((𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉)) → (𝑎𝑏 → (((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = inf({𝑎, 𝑏}, 𝑉, 𝑅) ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = sup({𝑎, 𝑏}, 𝑉, 𝑅)) → ((𝑐 = 𝑎𝑑 = 𝑏) ∨ (𝑐 = 𝑏𝑑 = 𝑎)))))))
10829adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑏𝑅𝑎 ∧ (𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉))) → inf({𝑎, 𝑏}, 𝑉, 𝑅) = if(𝑎𝑅𝑏, 𝑎, 𝑏))
109 sotric 5466 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑅 Or 𝑉 ∧ (𝑏𝑉𝑎𝑉)) → (𝑏𝑅𝑎 ↔ ¬ (𝑏 = 𝑎𝑎𝑅𝑏)))
110109ancom2s 649 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉)) → (𝑏𝑅𝑎 ↔ ¬ (𝑏 = 𝑎𝑎𝑅𝑏)))
111 ioran 981 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (¬ (𝑏 = 𝑎𝑎𝑅𝑏) ↔ (¬ 𝑏 = 𝑎 ∧ ¬ 𝑎𝑅𝑏))
112 iffalse 4434 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 𝑎𝑅𝑏 → if(𝑎𝑅𝑏, 𝑎, 𝑏) = 𝑏)
113111, 112simplbiim 508 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (¬ (𝑏 = 𝑎𝑎𝑅𝑏) → if(𝑎𝑅𝑏, 𝑎, 𝑏) = 𝑏)
114110, 113syl6bi 256 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉)) → (𝑏𝑅𝑎 → if(𝑎𝑅𝑏, 𝑎, 𝑏) = 𝑏))
115114impcom 411 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑏𝑅𝑎 ∧ (𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉))) → if(𝑎𝑅𝑏, 𝑎, 𝑏) = 𝑏)
116108, 115eqtrd 2833 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑏𝑅𝑎 ∧ (𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉))) → inf({𝑎, 𝑏}, 𝑉, 𝑅) = 𝑏)
117116eqeq2d 2809 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑏𝑅𝑎 ∧ (𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉))) → (inf({𝑐, 𝑑}, 𝑉, 𝑅) = inf({𝑎, 𝑏}, 𝑉, 𝑅) ↔ inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏))
118 iftrue 4431 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑏𝑅𝑎 → if(𝑏𝑅𝑎, 𝑎, 𝑏) = 𝑎)
11934, 118sylan9eqr 2855 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑏𝑅𝑎 ∧ (𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉))) → sup({𝑎, 𝑏}, 𝑉, 𝑅) = 𝑎)
120119eqeq2d 2809 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑏𝑅𝑎 ∧ (𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉))) → (sup({𝑐, 𝑑}, 𝑉, 𝑅) = sup({𝑎, 𝑏}, 𝑉, 𝑅) ↔ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎))
121117, 120anbi12d 633 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑏𝑅𝑎 ∧ (𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉))) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = inf({𝑎, 𝑏}, 𝑉, 𝑅) ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = sup({𝑎, 𝑏}, 𝑉, 𝑅)) ↔ (inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎)))
122121adantr 484 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑏𝑅𝑎 ∧ (𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉))) ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑)) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = inf({𝑎, 𝑏}, 𝑉, 𝑅) ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = sup({𝑎, 𝑏}, 𝑉, 𝑅)) ↔ (inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎)))
12354eqeq1d 2800 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑐𝑅𝑑 ∧ (𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑))) → (inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏𝑐 = 𝑏))
12466eqeq1d 2800 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑐𝑅𝑑 ∧ (𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑))) → (sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎𝑑 = 𝑎))
125123, 124anbi12d 633 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑐𝑅𝑑 ∧ (𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑))) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎) ↔ (𝑐 = 𝑏𝑑 = 𝑎)))
126125, 92syl6bi 256 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑐𝑅𝑑 ∧ (𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑))) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎) → ((𝑐 = 𝑎𝑑 = 𝑏) ∨ (𝑐 = 𝑏𝑑 = 𝑎))))
127126ex 416 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑐𝑅𝑑 → ((𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑)) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎) → ((𝑐 = 𝑎𝑑 = 𝑏) ∨ (𝑐 = 𝑏𝑑 = 𝑎)))))
128 eqneqall 2998 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑐 = 𝑑 → (𝑐𝑑 → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎) → ((𝑐 = 𝑎𝑑 = 𝑏) ∨ (𝑐 = 𝑏𝑑 = 𝑎)))))
129128adantld 494 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑐 = 𝑑 → (((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎) → ((𝑐 = 𝑎𝑑 = 𝑏) ∨ (𝑐 = 𝑏𝑑 = 𝑎)))))
130129adantld 494 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑐 = 𝑑 → ((𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑)) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎) → ((𝑐 = 𝑎𝑑 = 𝑏) ∨ (𝑐 = 𝑏𝑑 = 𝑎)))))
13184, 87syl6bi 256 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑)) → (𝑑𝑅𝑐 → if(𝑐𝑅𝑑, 𝑐, 𝑑) = 𝑑))
132131impcom 411 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑑𝑅𝑐 ∧ (𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑))) → if(𝑐𝑅𝑑, 𝑐, 𝑑) = 𝑑)
13375, 132eqtrd 2833 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑑𝑅𝑐 ∧ (𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑))) → inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑑)
134133eqeq1d 2800 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑑𝑅𝑐 ∧ (𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑))) → (inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏𝑑 = 𝑏))
13578eqeq1d 2800 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑑𝑅𝑐 ∧ (𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑))) → (sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎𝑐 = 𝑎))
136134, 135anbi12d 633 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑑𝑅𝑐 ∧ (𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑))) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎) ↔ (𝑑 = 𝑏𝑐 = 𝑎)))
13769ancoms 462 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑑 = 𝑏𝑐 = 𝑎) → ((𝑐 = 𝑎𝑑 = 𝑏) ∨ (𝑐 = 𝑏𝑑 = 𝑎)))
138136, 137syl6bi 256 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑑𝑅𝑐 ∧ (𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑))) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎) → ((𝑐 = 𝑎𝑑 = 𝑏) ∨ (𝑐 = 𝑏𝑑 = 𝑎))))
139138ex 416 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑑𝑅𝑐 → ((𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑)) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎) → ((𝑐 = 𝑎𝑑 = 𝑏) ∨ (𝑐 = 𝑏𝑑 = 𝑎)))))
140127, 130, 1393jaoi 1424 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑐𝑅𝑑𝑐 = 𝑑𝑑𝑅𝑐) → ((𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑)) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎) → ((𝑐 = 𝑎𝑑 = 𝑏) ∨ (𝑐 = 𝑏𝑑 = 𝑎)))))
14147, 140mpcom 38 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑)) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎) → ((𝑐 = 𝑎𝑑 = 𝑏) ∨ (𝑐 = 𝑏𝑑 = 𝑎))))
142141ex 416 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑅 Or 𝑉 → (((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎) → ((𝑐 = 𝑎𝑑 = 𝑏) ∨ (𝑐 = 𝑏𝑑 = 𝑎)))))
143142ad2antrl 727 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑏𝑅𝑎 ∧ (𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉))) → (((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎) → ((𝑐 = 𝑎𝑑 = 𝑏) ∨ (𝑐 = 𝑏𝑑 = 𝑎)))))
144143imp 410 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑏𝑅𝑎 ∧ (𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉))) ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑)) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎) → ((𝑐 = 𝑎𝑑 = 𝑏) ∨ (𝑐 = 𝑏𝑑 = 𝑎))))
145122, 144sylbid 243 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑏𝑅𝑎 ∧ (𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉))) ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑)) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = inf({𝑎, 𝑏}, 𝑉, 𝑅) ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = sup({𝑎, 𝑏}, 𝑉, 𝑅)) → ((𝑐 = 𝑎𝑑 = 𝑏) ∨ (𝑐 = 𝑏𝑑 = 𝑎))))
146145ex 416 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑏𝑅𝑎 ∧ (𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉))) → (((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = inf({𝑎, 𝑏}, 𝑉, 𝑅) ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = sup({𝑎, 𝑏}, 𝑉, 𝑅)) → ((𝑐 = 𝑎𝑑 = 𝑏) ∨ (𝑐 = 𝑏𝑑 = 𝑎)))))
147146a1d 25 . . . . . . . . . . . . . . . . . . . . 21 ((𝑏𝑅𝑎 ∧ (𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉))) → (𝑎𝑏 → (((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = inf({𝑎, 𝑏}, 𝑉, 𝑅) ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = sup({𝑎, 𝑏}, 𝑉, 𝑅)) → ((𝑐 = 𝑎𝑑 = 𝑏) ∨ (𝑐 = 𝑏𝑑 = 𝑎))))))
148147ex 416 . . . . . . . . . . . . . . . . . . . 20 (𝑏𝑅𝑎 → ((𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉)) → (𝑎𝑏 → (((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = inf({𝑎, 𝑏}, 𝑉, 𝑅) ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = sup({𝑎, 𝑏}, 𝑉, 𝑅)) → ((𝑐 = 𝑎𝑑 = 𝑏) ∨ (𝑐 = 𝑏𝑑 = 𝑎)))))))
149105, 107, 1483jaoi 1424 . . . . . . . . . . . . . . . . . . 19 ((𝑎𝑅𝑏𝑎 = 𝑏𝑏𝑅𝑎) → ((𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉)) → (𝑎𝑏 → (((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = inf({𝑎, 𝑏}, 𝑉, 𝑅) ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = sup({𝑎, 𝑏}, 𝑉, 𝑅)) → ((𝑐 = 𝑎𝑑 = 𝑏) ∨ (𝑐 = 𝑏𝑑 = 𝑎)))))))
15027, 149mpcom 38 . . . . . . . . . . . . . . . . . 18 ((𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉)) → (𝑎𝑏 → (((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = inf({𝑎, 𝑏}, 𝑉, 𝑅) ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = sup({𝑎, 𝑏}, 𝑉, 𝑅)) → ((𝑐 = 𝑎𝑑 = 𝑏) ∨ (𝑐 = 𝑏𝑑 = 𝑎))))))
151150adantld 494 . . . . . . . . . . . . . . . . 17 ((𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉)) → ((𝑊 = {𝑎, 𝑏} ∧ 𝑎𝑏) → (((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = inf({𝑎, 𝑏}, 𝑉, 𝑅) ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = sup({𝑎, 𝑏}, 𝑉, 𝑅)) → ((𝑐 = 𝑎𝑑 = 𝑏) ∨ (𝑐 = 𝑏𝑑 = 𝑎))))))
152151imp 410 . . . . . . . . . . . . . . . 16 (((𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑊 = {𝑎, 𝑏} ∧ 𝑎𝑏)) → (((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = inf({𝑎, 𝑏}, 𝑉, 𝑅) ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = sup({𝑎, 𝑏}, 𝑉, 𝑅)) → ((𝑐 = 𝑎𝑑 = 𝑏) ∨ (𝑐 = 𝑏𝑑 = 𝑎)))))
153152expdimp 456 . . . . . . . . . . . . . . 15 ((((𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑊 = {𝑎, 𝑏} ∧ 𝑎𝑏)) ∧ (𝑐𝑉𝑑𝑉)) → (𝑐𝑑 → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = inf({𝑎, 𝑏}, 𝑉, 𝑅) ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = sup({𝑎, 𝑏}, 𝑉, 𝑅)) → ((𝑐 = 𝑎𝑑 = 𝑏) ∨ (𝑐 = 𝑏𝑑 = 𝑎)))))
154153adantld 494 . . . . . . . . . . . . . 14 ((((𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑊 = {𝑎, 𝑏} ∧ 𝑎𝑏)) ∧ (𝑐𝑉𝑑𝑉)) → ((𝑍 = {𝑐, 𝑑} ∧ 𝑐𝑑) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = inf({𝑎, 𝑏}, 𝑉, 𝑅) ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = sup({𝑎, 𝑏}, 𝑉, 𝑅)) → ((𝑐 = 𝑎𝑑 = 𝑏) ∨ (𝑐 = 𝑏𝑑 = 𝑎)))))
155154imp 410 . . . . . . . . . . . . 13 (((((𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑊 = {𝑎, 𝑏} ∧ 𝑎𝑏)) ∧ (𝑐𝑉𝑑𝑉)) ∧ (𝑍 = {𝑐, 𝑑} ∧ 𝑐𝑑)) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = inf({𝑎, 𝑏}, 𝑉, 𝑅) ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = sup({𝑎, 𝑏}, 𝑉, 𝑅)) → ((𝑐 = 𝑎𝑑 = 𝑏) ∨ (𝑐 = 𝑏𝑑 = 𝑎))))
156 vex 3444 . . . . . . . . . . . . . 14 𝑐 ∈ V
157 vex 3444 . . . . . . . . . . . . . 14 𝑑 ∈ V
158 vex 3444 . . . . . . . . . . . . . 14 𝑎 ∈ V
159 vex 3444 . . . . . . . . . . . . . 14 𝑏 ∈ V
160156, 157, 158, 159preq12b 4741 . . . . . . . . . . . . 13 ({𝑐, 𝑑} = {𝑎, 𝑏} ↔ ((𝑐 = 𝑎𝑑 = 𝑏) ∨ (𝑐 = 𝑏𝑑 = 𝑎)))
161155, 160syl6ibr 255 . . . . . . . . . . . 12 (((((𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑊 = {𝑎, 𝑏} ∧ 𝑎𝑏)) ∧ (𝑐𝑉𝑑𝑉)) ∧ (𝑍 = {𝑐, 𝑑} ∧ 𝑐𝑑)) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = inf({𝑎, 𝑏}, 𝑉, 𝑅) ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = sup({𝑎, 𝑏}, 𝑉, 𝑅)) → {𝑐, 𝑑} = {𝑎, 𝑏}))
16226, 161sylbid 243 . . . . . . . . . . 11 (((((𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑊 = {𝑎, 𝑏} ∧ 𝑎𝑏)) ∧ (𝑐𝑉𝑑𝑉)) ∧ (𝑍 = {𝑐, 𝑑} ∧ 𝑐𝑑)) → (⟨inf({𝑐, 𝑑}, 𝑉, 𝑅), sup({𝑐, 𝑑}, 𝑉, 𝑅)⟩ = ⟨inf({𝑎, 𝑏}, 𝑉, 𝑅), sup({𝑎, 𝑏}, 𝑉, 𝑅)⟩ → {𝑐, 𝑑} = {𝑎, 𝑏}))
163 infeq1 8927 . . . . . . . . . . . . . . . . . . . 20 (𝑍 = {𝑐, 𝑑} → inf(𝑍, 𝑉, 𝑅) = inf({𝑐, 𝑑}, 𝑉, 𝑅))
164 supeq1 8896 . . . . . . . . . . . . . . . . . . . 20 (𝑍 = {𝑐, 𝑑} → sup(𝑍, 𝑉, 𝑅) = sup({𝑐, 𝑑}, 𝑉, 𝑅))
165163, 164opeq12d 4774 . . . . . . . . . . . . . . . . . . 19 (𝑍 = {𝑐, 𝑑} → ⟨inf(𝑍, 𝑉, 𝑅), sup(𝑍, 𝑉, 𝑅)⟩ = ⟨inf({𝑐, 𝑑}, 𝑉, 𝑅), sup({𝑐, 𝑑}, 𝑉, 𝑅)⟩)
166 infeq1 8927 . . . . . . . . . . . . . . . . . . . 20 (𝑊 = {𝑎, 𝑏} → inf(𝑊, 𝑉, 𝑅) = inf({𝑎, 𝑏}, 𝑉, 𝑅))
167 supeq1 8896 . . . . . . . . . . . . . . . . . . . 20 (𝑊 = {𝑎, 𝑏} → sup(𝑊, 𝑉, 𝑅) = sup({𝑎, 𝑏}, 𝑉, 𝑅))
168166, 167opeq12d 4774 . . . . . . . . . . . . . . . . . . 19 (𝑊 = {𝑎, 𝑏} → ⟨inf(𝑊, 𝑉, 𝑅), sup(𝑊, 𝑉, 𝑅)⟩ = ⟨inf({𝑎, 𝑏}, 𝑉, 𝑅), sup({𝑎, 𝑏}, 𝑉, 𝑅)⟩)
169165, 168eqeqan12rd 2817 . . . . . . . . . . . . . . . . . 18 ((𝑊 = {𝑎, 𝑏} ∧ 𝑍 = {𝑐, 𝑑}) → (⟨inf(𝑍, 𝑉, 𝑅), sup(𝑍, 𝑉, 𝑅)⟩ = ⟨inf(𝑊, 𝑉, 𝑅), sup(𝑊, 𝑉, 𝑅)⟩ ↔ ⟨inf({𝑐, 𝑑}, 𝑉, 𝑅), sup({𝑐, 𝑑}, 𝑉, 𝑅)⟩ = ⟨inf({𝑎, 𝑏}, 𝑉, 𝑅), sup({𝑎, 𝑏}, 𝑉, 𝑅)⟩))
170 eqeq12 2812 . . . . . . . . . . . . . . . . . . 19 ((𝑍 = {𝑐, 𝑑} ∧ 𝑊 = {𝑎, 𝑏}) → (𝑍 = 𝑊 ↔ {𝑐, 𝑑} = {𝑎, 𝑏}))
171170ancoms 462 . . . . . . . . . . . . . . . . . 18 ((𝑊 = {𝑎, 𝑏} ∧ 𝑍 = {𝑐, 𝑑}) → (𝑍 = 𝑊 ↔ {𝑐, 𝑑} = {𝑎, 𝑏}))
172169, 171imbi12d 348 . . . . . . . . . . . . . . . . 17 ((𝑊 = {𝑎, 𝑏} ∧ 𝑍 = {𝑐, 𝑑}) → ((⟨inf(𝑍, 𝑉, 𝑅), sup(𝑍, 𝑉, 𝑅)⟩ = ⟨inf(𝑊, 𝑉, 𝑅), sup(𝑊, 𝑉, 𝑅)⟩ → 𝑍 = 𝑊) ↔ (⟨inf({𝑐, 𝑑}, 𝑉, 𝑅), sup({𝑐, 𝑑}, 𝑉, 𝑅)⟩ = ⟨inf({𝑎, 𝑏}, 𝑉, 𝑅), sup({𝑎, 𝑏}, 𝑉, 𝑅)⟩ → {𝑐, 𝑑} = {𝑎, 𝑏})))
173172ex 416 . . . . . . . . . . . . . . . 16 (𝑊 = {𝑎, 𝑏} → (𝑍 = {𝑐, 𝑑} → ((⟨inf(𝑍, 𝑉, 𝑅), sup(𝑍, 𝑉, 𝑅)⟩ = ⟨inf(𝑊, 𝑉, 𝑅), sup(𝑊, 𝑉, 𝑅)⟩ → 𝑍 = 𝑊) ↔ (⟨inf({𝑐, 𝑑}, 𝑉, 𝑅), sup({𝑐, 𝑑}, 𝑉, 𝑅)⟩ = ⟨inf({𝑎, 𝑏}, 𝑉, 𝑅), sup({𝑎, 𝑏}, 𝑉, 𝑅)⟩ → {𝑐, 𝑑} = {𝑎, 𝑏}))))
174173ad2antrl 727 . . . . . . . . . . . . . . 15 (((𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑊 = {𝑎, 𝑏} ∧ 𝑎𝑏)) → (𝑍 = {𝑐, 𝑑} → ((⟨inf(𝑍, 𝑉, 𝑅), sup(𝑍, 𝑉, 𝑅)⟩ = ⟨inf(𝑊, 𝑉, 𝑅), sup(𝑊, 𝑉, 𝑅)⟩ → 𝑍 = 𝑊) ↔ (⟨inf({𝑐, 𝑑}, 𝑉, 𝑅), sup({𝑐, 𝑑}, 𝑉, 𝑅)⟩ = ⟨inf({𝑎, 𝑏}, 𝑉, 𝑅), sup({𝑎, 𝑏}, 𝑉, 𝑅)⟩ → {𝑐, 𝑑} = {𝑎, 𝑏}))))
175174adantr 484 . . . . . . . . . . . . . 14 ((((𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑊 = {𝑎, 𝑏} ∧ 𝑎𝑏)) ∧ (𝑐𝑉𝑑𝑉)) → (𝑍 = {𝑐, 𝑑} → ((⟨inf(𝑍, 𝑉, 𝑅), sup(𝑍, 𝑉, 𝑅)⟩ = ⟨inf(𝑊, 𝑉, 𝑅), sup(𝑊, 𝑉, 𝑅)⟩ → 𝑍 = 𝑊) ↔ (⟨inf({𝑐, 𝑑}, 𝑉, 𝑅), sup({𝑐, 𝑑}, 𝑉, 𝑅)⟩ = ⟨inf({𝑎, 𝑏}, 𝑉, 𝑅), sup({𝑎, 𝑏}, 𝑉, 𝑅)⟩ → {𝑐, 𝑑} = {𝑎, 𝑏}))))
176175com12 32 . . . . . . . . . . . . 13 (𝑍 = {𝑐, 𝑑} → ((((𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑊 = {𝑎, 𝑏} ∧ 𝑎𝑏)) ∧ (𝑐𝑉𝑑𝑉)) → ((⟨inf(𝑍, 𝑉, 𝑅), sup(𝑍, 𝑉, 𝑅)⟩ = ⟨inf(𝑊, 𝑉, 𝑅), sup(𝑊, 𝑉, 𝑅)⟩ → 𝑍 = 𝑊) ↔ (⟨inf({𝑐, 𝑑}, 𝑉, 𝑅), sup({𝑐, 𝑑}, 𝑉, 𝑅)⟩ = ⟨inf({𝑎, 𝑏}, 𝑉, 𝑅), sup({𝑎, 𝑏}, 𝑉, 𝑅)⟩ → {𝑐, 𝑑} = {𝑎, 𝑏}))))
177176adantr 484 . . . . . . . . . . . 12 ((𝑍 = {𝑐, 𝑑} ∧ 𝑐𝑑) → ((((𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑊 = {𝑎, 𝑏} ∧ 𝑎𝑏)) ∧ (𝑐𝑉𝑑𝑉)) → ((⟨inf(𝑍, 𝑉, 𝑅), sup(𝑍, 𝑉, 𝑅)⟩ = ⟨inf(𝑊, 𝑉, 𝑅), sup(𝑊, 𝑉, 𝑅)⟩ → 𝑍 = 𝑊) ↔ (⟨inf({𝑐, 𝑑}, 𝑉, 𝑅), sup({𝑐, 𝑑}, 𝑉, 𝑅)⟩ = ⟨inf({𝑎, 𝑏}, 𝑉, 𝑅), sup({𝑎, 𝑏}, 𝑉, 𝑅)⟩ → {𝑐, 𝑑} = {𝑎, 𝑏}))))
178177impcom 411 . . . . . . . . . . 11 (((((𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑊 = {𝑎, 𝑏} ∧ 𝑎𝑏)) ∧ (𝑐𝑉𝑑𝑉)) ∧ (𝑍 = {𝑐, 𝑑} ∧ 𝑐𝑑)) → ((⟨inf(𝑍, 𝑉, 𝑅), sup(𝑍, 𝑉, 𝑅)⟩ = ⟨inf(𝑊, 𝑉, 𝑅), sup(𝑊, 𝑉, 𝑅)⟩ → 𝑍 = 𝑊) ↔ (⟨inf({𝑐, 𝑑}, 𝑉, 𝑅), sup({𝑐, 𝑑}, 𝑉, 𝑅)⟩ = ⟨inf({𝑎, 𝑏}, 𝑉, 𝑅), sup({𝑎, 𝑏}, 𝑉, 𝑅)⟩ → {𝑐, 𝑑} = {𝑎, 𝑏})))
179162, 178mpbird 260 . . . . . . . . . 10 (((((𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑊 = {𝑎, 𝑏} ∧ 𝑎𝑏)) ∧ (𝑐𝑉𝑑𝑉)) ∧ (𝑍 = {𝑐, 𝑑} ∧ 𝑐𝑑)) → (⟨inf(𝑍, 𝑉, 𝑅), sup(𝑍, 𝑉, 𝑅)⟩ = ⟨inf(𝑊, 𝑉, 𝑅), sup(𝑊, 𝑉, 𝑅)⟩ → 𝑍 = 𝑊))
180179ex 416 . . . . . . . . 9 ((((𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑊 = {𝑎, 𝑏} ∧ 𝑎𝑏)) ∧ (𝑐𝑉𝑑𝑉)) → ((𝑍 = {𝑐, 𝑑} ∧ 𝑐𝑑) → (⟨inf(𝑍, 𝑉, 𝑅), sup(𝑍, 𝑉, 𝑅)⟩ = ⟨inf(𝑊, 𝑉, 𝑅), sup(𝑊, 𝑉, 𝑅)⟩ → 𝑍 = 𝑊)))
181180rexlimdvva 3253 . . . . . . . 8 (((𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑊 = {𝑎, 𝑏} ∧ 𝑎𝑏)) → (∃𝑐𝑉𝑑𝑉 (𝑍 = {𝑐, 𝑑} ∧ 𝑐𝑑) → (⟨inf(𝑍, 𝑉, 𝑅), sup(𝑍, 𝑉, 𝑅)⟩ = ⟨inf(𝑊, 𝑉, 𝑅), sup(𝑊, 𝑉, 𝑅)⟩ → 𝑍 = 𝑊)))
182181ex 416 . . . . . . 7 ((𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉)) → ((𝑊 = {𝑎, 𝑏} ∧ 𝑎𝑏) → (∃𝑐𝑉𝑑𝑉 (𝑍 = {𝑐, 𝑑} ∧ 𝑐𝑑) → (⟨inf(𝑍, 𝑉, 𝑅), sup(𝑍, 𝑉, 𝑅)⟩ = ⟨inf(𝑊, 𝑉, 𝑅), sup(𝑊, 𝑉, 𝑅)⟩ → 𝑍 = 𝑊))))
183182rexlimdvva 3253 . . . . . 6 (𝑅 Or 𝑉 → (∃𝑎𝑉𝑏𝑉 (𝑊 = {𝑎, 𝑏} ∧ 𝑎𝑏) → (∃𝑐𝑉𝑑𝑉 (𝑍 = {𝑐, 𝑑} ∧ 𝑐𝑑) → (⟨inf(𝑍, 𝑉, 𝑅), sup(𝑍, 𝑉, 𝑅)⟩ = ⟨inf(𝑊, 𝑉, 𝑅), sup(𝑊, 𝑉, 𝑅)⟩ → 𝑍 = 𝑊))))
184183com13 88 . . . . 5 (∃𝑐𝑉𝑑𝑉 (𝑍 = {𝑐, 𝑑} ∧ 𝑐𝑑) → (∃𝑎𝑉𝑏𝑉 (𝑊 = {𝑎, 𝑏} ∧ 𝑎𝑏) → (𝑅 Or 𝑉 → (⟨inf(𝑍, 𝑉, 𝑅), sup(𝑍, 𝑉, 𝑅)⟩ = ⟨inf(𝑊, 𝑉, 𝑅), sup(𝑊, 𝑉, 𝑅)⟩ → 𝑍 = 𝑊))))
18519, 184syl5bi 245 . . . 4 (∃𝑐𝑉𝑑𝑉 (𝑍 = {𝑐, 𝑑} ∧ 𝑐𝑑) → (𝑊𝑃 → (𝑅 Or 𝑉 → (⟨inf(𝑍, 𝑉, 𝑅), sup(𝑍, 𝑉, 𝑅)⟩ = ⟨inf(𝑊, 𝑉, 𝑅), sup(𝑊, 𝑉, 𝑅)⟩ → 𝑍 = 𝑊))))
18618, 185sylbi 220 . . 3 (𝑍𝑃 → (𝑊𝑃 → (𝑅 Or 𝑉 → (⟨inf(𝑍, 𝑉, 𝑅), sup(𝑍, 𝑉, 𝑅)⟩ = ⟨inf(𝑊, 𝑉, 𝑅), sup(𝑊, 𝑉, 𝑅)⟩ → 𝑍 = 𝑊))))
1871863imp31 1109 . 2 ((𝑅 Or 𝑉𝑊𝑃𝑍𝑃) → (⟨inf(𝑍, 𝑉, 𝑅), sup(𝑍, 𝑉, 𝑅)⟩ = ⟨inf(𝑊, 𝑉, 𝑅), sup(𝑊, 𝑉, 𝑅)⟩ → 𝑍 = 𝑊))
18816, 187sylbid 243 1 ((𝑅 Or 𝑉𝑊𝑃𝑍𝑃) → ((𝐹𝑍) = (𝐹𝑊) → 𝑍 = 𝑊))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   ∨ w3o 1083   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∃wrex 3107  {crab 3110  Vcvv 3441   ∩ cin 3880  ifcif 4425  𝒫 cpw 4497  {cpr 4527  ⟨cop 4531   class class class wbr 5031   ↦ cmpt 5111   Or wor 5438   × cxp 5518  ‘cfv 6325  supcsup 8891  infcinf 8892  2c2 11683  ♯chash 13689 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4840  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-pred 6117  df-ord 6163  df-on 6164  df-lim 6165  df-suc 6166  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-riota 7094  df-ov 7139  df-oprab 7140  df-mpo 7141  df-om 7564  df-1st 7674  df-2nd 7675  df-wrecs 7933  df-recs 7994  df-rdg 8032  df-1o 8088  df-2o 8089  df-oadd 8092  df-er 8275  df-en 8496  df-dom 8497  df-sdom 8498  df-fin 8499  df-sup 8893  df-inf 8894  df-dju 9317  df-card 9355  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11629  df-2 11691  df-n0 11889  df-z 11973  df-uz 12235  df-fz 12889  df-hash 13690 This theorem is referenced by:  prproropf1o  44067
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