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Theorem prproropf1olem4 47380
Description: Lemma 4 for prproropf1o 47381. (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 9545 . . . . 5 (𝑝 = 𝑍 → inf(𝑝, 𝑉, 𝑅) = inf(𝑍, 𝑉, 𝑅))
3 supeq1 9514 . . . . 5 (𝑝 = 𝑍 → sup(𝑝, 𝑉, 𝑅) = sup(𝑍, 𝑉, 𝑅))
42, 3opeq12d 4905 . . . 4 (𝑝 = 𝑍 → ⟨inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)⟩ = ⟨inf(𝑍, 𝑉, 𝑅), sup(𝑍, 𝑉, 𝑅)⟩)
5 simp3 1138 . . . 4 ((𝑅 Or 𝑉𝑊𝑃𝑍𝑃) → 𝑍𝑃)
6 opex 5484 . . . . 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 7052 . . 3 ((𝑅 Or 𝑉𝑊𝑃𝑍𝑃) → (𝐹𝑍) = ⟨inf(𝑍, 𝑉, 𝑅), sup(𝑍, 𝑉, 𝑅)⟩)
9 infeq1 9545 . . . . 5 (𝑝 = 𝑊 → inf(𝑝, 𝑉, 𝑅) = inf(𝑊, 𝑉, 𝑅))
10 supeq1 9514 . . . . 5 (𝑝 = 𝑊 → sup(𝑝, 𝑉, 𝑅) = sup(𝑊, 𝑉, 𝑅))
119, 10opeq12d 4905 . . . 4 (𝑝 = 𝑊 → ⟨inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)⟩ = ⟨inf(𝑊, 𝑉, 𝑅), sup(𝑊, 𝑉, 𝑅)⟩)
12 simp2 1137 . . . 4 ((𝑅 Or 𝑉𝑊𝑃𝑍𝑃) → 𝑊𝑃)
13 opex 5484 . . . . 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 7052 . . 3 ((𝑅 Or 𝑉𝑊𝑃𝑍𝑃) → (𝐹𝑊) = ⟨inf(𝑊, 𝑉, 𝑅), sup(𝑊, 𝑉, 𝑅)⟩)
168, 15eqeq12d 2756 . 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 47375 . . . 4 (𝑍𝑃 ↔ ∃𝑐𝑉𝑑𝑉 (𝑍 = {𝑐, 𝑑} ∧ 𝑐𝑑))
1917prpair 47375 . . . . 5 (𝑊𝑃 ↔ ∃𝑎𝑉𝑏𝑉 (𝑊 = {𝑎, 𝑏} ∧ 𝑎𝑏))
20 id 22 . . . . . . . . . . . . . . . 16 (𝑅 Or 𝑉𝑅 Or 𝑉)
2120infexd 9552 . . . . . . . . . . . . . . 15 (𝑅 Or 𝑉 → inf({𝑐, 𝑑}, 𝑉, 𝑅) ∈ V)
2220supexd 9522 . . . . . . . . . . . . . . 15 (𝑅 Or 𝑉 → sup({𝑐, 𝑑}, 𝑉, 𝑅) ∈ V)
2321, 22jca 511 . . . . . . . . . . . . . 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 5497 . . . . . . . . . . . . 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 5634 . . . . . . . . . . . . . . . . . . 19 ((𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉)) → (𝑎𝑅𝑏𝑎 = 𝑏𝑏𝑅𝑎))
28 infpr 9572 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑅 Or 𝑉𝑎𝑉𝑏𝑉) → inf({𝑎, 𝑏}, 𝑉, 𝑅) = if(𝑎𝑅𝑏, 𝑎, 𝑏))
29283expb 1120 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉)) → inf({𝑎, 𝑏}, 𝑉, 𝑅) = if(𝑎𝑅𝑏, 𝑎, 𝑏))
30 iftrue 4554 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑎𝑅𝑏 → if(𝑎𝑅𝑏, 𝑎, 𝑏) = 𝑎)
3129, 30sylan9eqr 2802 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑎𝑅𝑏 ∧ (𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉))) → inf({𝑎, 𝑏}, 𝑉, 𝑅) = 𝑎)
3231eqeq2d 2751 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑎𝑅𝑏 ∧ (𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉))) → (inf({𝑐, 𝑑}, 𝑉, 𝑅) = inf({𝑎, 𝑏}, 𝑉, 𝑅) ↔ inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎))
33 suppr 9540 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑅 Or 𝑉𝑎𝑉𝑏𝑉) → sup({𝑎, 𝑏}, 𝑉, 𝑅) = if(𝑏𝑅𝑎, 𝑎, 𝑏))
34333expb 1120 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉)) → sup({𝑎, 𝑏}, 𝑉, 𝑅) = if(𝑏𝑅𝑎, 𝑎, 𝑏))
3534adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑎𝑅𝑏 ∧ (𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉))) → sup({𝑎, 𝑏}, 𝑉, 𝑅) = if(𝑏𝑅𝑎, 𝑎, 𝑏))
36 sotric 5637 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉)) → (𝑎𝑅𝑏 ↔ ¬ (𝑎 = 𝑏𝑏𝑅𝑎)))
37 ioran 984 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (¬ (𝑎 = 𝑏𝑏𝑅𝑎) ↔ (¬ 𝑎 = 𝑏 ∧ ¬ 𝑏𝑅𝑎))
38 iffalse 4557 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 𝑏𝑅𝑎 → if(𝑏𝑅𝑎, 𝑎, 𝑏) = 𝑏)
3937, 38simplbiim 504 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (¬ (𝑎 = 𝑏𝑏𝑅𝑎) → if(𝑏𝑅𝑎, 𝑎, 𝑏) = 𝑏)
4036, 39biimtrdi 253 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉)) → (𝑎𝑅𝑏 → if(𝑏𝑅𝑎, 𝑎, 𝑏) = 𝑏))
4140impcom 407 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑎𝑅𝑏 ∧ (𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉))) → if(𝑏𝑅𝑎, 𝑎, 𝑏) = 𝑏)
4235, 41eqtrd 2780 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑎𝑅𝑏 ∧ (𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉))) → sup({𝑎, 𝑏}, 𝑉, 𝑅) = 𝑏)
4342eqeq2d 2751 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑎𝑅𝑏 ∧ (𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉))) → (sup({𝑐, 𝑑}, 𝑉, 𝑅) = sup({𝑎, 𝑏}, 𝑉, 𝑅) ↔ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏))
4432, 43anbi12d 631 . . . . . . . . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . . . . . . . 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 5634 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑅 Or 𝑉 ∧ (𝑐𝑉𝑑𝑉)) → (𝑐𝑅𝑑𝑐 = 𝑑𝑑𝑅𝑐))
4746adantrr 716 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑)) → (𝑐𝑅𝑑𝑐 = 𝑑𝑑𝑅𝑐))
48 simpl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 9572 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑅 Or 𝑉𝑐𝑉𝑑𝑉) → inf({𝑐, 𝑑}, 𝑉, 𝑅) = if(𝑐𝑅𝑑, 𝑐, 𝑑))
5248, 49, 50, 51syl3anc 1371 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑)) → inf({𝑐, 𝑑}, 𝑉, 𝑅) = if(𝑐𝑅𝑑, 𝑐, 𝑑))
53 iftrue 4554 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑐𝑅𝑑 → if(𝑐𝑅𝑑, 𝑐, 𝑑) = 𝑐)
5452, 53sylan9eqr 2802 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑐𝑅𝑑 ∧ (𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑))) → inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑐)
5554eqeq1d 2742 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑐𝑅𝑑 ∧ (𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑))) → (inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎𝑐 = 𝑎))
56 suppr 9540 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑅 Or 𝑉𝑐𝑉𝑑𝑉) → sup({𝑐, 𝑑}, 𝑉, 𝑅) = if(𝑑𝑅𝑐, 𝑐, 𝑑))
5748, 49, 50, 56syl3anc 1371 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑)) → sup({𝑐, 𝑑}, 𝑉, 𝑅) = if(𝑑𝑅𝑐, 𝑐, 𝑑))
5857adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑐𝑅𝑑 ∧ (𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑))) → sup({𝑐, 𝑑}, 𝑉, 𝑅) = if(𝑑𝑅𝑐, 𝑐, 𝑑))
59 sotric 5637 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑅 Or 𝑉 ∧ (𝑐𝑉𝑑𝑉)) → (𝑐𝑅𝑑 ↔ ¬ (𝑐 = 𝑑𝑑𝑅𝑐)))
6059adantrr 716 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑)) → (𝑐𝑅𝑑 ↔ ¬ (𝑐 = 𝑑𝑑𝑅𝑐)))
61 ioran 984 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (¬ (𝑐 = 𝑑𝑑𝑅𝑐) ↔ (¬ 𝑐 = 𝑑 ∧ ¬ 𝑑𝑅𝑐))
62 iffalse 4557 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 𝑑𝑅𝑐 → if(𝑑𝑅𝑐, 𝑐, 𝑑) = 𝑑)
6361, 62simplbiim 504 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (¬ (𝑐 = 𝑑𝑑𝑅𝑐) → if(𝑑𝑅𝑐, 𝑐, 𝑑) = 𝑑)
6460, 63biimtrdi 253 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑)) → (𝑐𝑅𝑑 → if(𝑑𝑅𝑐, 𝑐, 𝑑) = 𝑑))
6564impcom 407 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑐𝑅𝑑 ∧ (𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑))) → if(𝑑𝑅𝑐, 𝑐, 𝑑) = 𝑑)
6658, 65eqtrd 2780 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑐𝑅𝑑 ∧ (𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑))) → sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑑)
6766eqeq1d 2742 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑐𝑅𝑑 ∧ (𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑))) → (sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏𝑑 = 𝑏))
6855, 67anbi12d 631 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑐𝑅𝑑 ∧ (𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑))) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏) ↔ (𝑐 = 𝑎𝑑 = 𝑏)))
69 orc 866 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑐 = 𝑎𝑑 = 𝑏) → ((𝑐 = 𝑎𝑑 = 𝑏) ∨ (𝑐 = 𝑏𝑑 = 𝑎)))
7068, 69biimtrdi 253 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 412 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2957 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑐 = 𝑑 → (𝑐𝑑 → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏) → ((𝑐 = 𝑎𝑑 = 𝑏) ∨ (𝑐 = 𝑏𝑑 = 𝑎)))))
7372adantld 490 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑐 = 𝑑 → (((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏) → ((𝑐 = 𝑎𝑑 = 𝑏) ∨ (𝑐 = 𝑏𝑑 = 𝑎)))))
7473adantld 490 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑑𝑅𝑐 ∧ (𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑))) → inf({𝑐, 𝑑}, 𝑉, 𝑅) = if(𝑐𝑅𝑑, 𝑐, 𝑑))
7675eqeq1d 2742 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑑𝑅𝑐 ∧ (𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑))) → (inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎 ↔ if(𝑐𝑅𝑑, 𝑐, 𝑑) = 𝑎))
77 iftrue 4554 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑑𝑅𝑐 → if(𝑑𝑅𝑐, 𝑐, 𝑑) = 𝑐)
7857, 77sylan9eqr 2802 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑑𝑅𝑐 ∧ (𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑))) → sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑐)
7978eqeq1d 2742 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑑𝑅𝑐 ∧ (𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑))) → (sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏𝑐 = 𝑏))
8076, 79anbi12d 631 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑) → (𝑐𝑉𝑑𝑉))
8281ancomd 461 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑) → (𝑑𝑉𝑐𝑉))
83 sotric 5637 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑅 Or 𝑉 ∧ (𝑑𝑉𝑐𝑉)) → (𝑑𝑅𝑐 ↔ ¬ (𝑑 = 𝑐𝑐𝑅𝑑)))
8482, 83sylan2 592 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑)) → (𝑑𝑅𝑐 ↔ ¬ (𝑑 = 𝑐𝑐𝑅𝑑)))
85 ioran 984 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (¬ (𝑑 = 𝑐𝑐𝑅𝑑) ↔ (¬ 𝑑 = 𝑐 ∧ ¬ 𝑐𝑅𝑑))
86 iffalse 4557 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 𝑐𝑅𝑑 → if(𝑐𝑅𝑑, 𝑐, 𝑑) = 𝑑)
8785, 86simplbiim 504 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (¬ (𝑑 = 𝑐𝑐𝑅𝑑) → if(𝑐𝑅𝑑, 𝑐, 𝑑) = 𝑑)
8887eqeq1d 2742 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (¬ (𝑑 = 𝑐𝑐𝑅𝑑) → (if(𝑐𝑅𝑑, 𝑐, 𝑑) = 𝑎𝑑 = 𝑎))
8984, 88biimtrdi 253 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑)) → (𝑑𝑅𝑐 → (if(𝑐𝑅𝑑, 𝑐, 𝑑) = 𝑎𝑑 = 𝑎)))
9089impcom 407 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑑𝑅𝑐 ∧ (𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑))) → (if(𝑐𝑅𝑑, 𝑐, 𝑑) = 𝑎𝑑 = 𝑎))
9190anbi1d 630 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑑𝑅𝑐 ∧ (𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑))) → ((if(𝑐𝑅𝑑, 𝑐, 𝑑) = 𝑎𝑐 = 𝑏) ↔ (𝑑 = 𝑎𝑐 = 𝑏)))
92 olc 867 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑐 = 𝑏𝑑 = 𝑎) → ((𝑐 = 𝑎𝑑 = 𝑏) ∨ (𝑐 = 𝑏𝑑 = 𝑎)))
9392ancoms 458 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑑 = 𝑎𝑐 = 𝑏) → ((𝑐 = 𝑎𝑑 = 𝑏) ∨ (𝑐 = 𝑏𝑑 = 𝑎)))
9491, 93biimtrdi 253 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑑𝑅𝑐 ∧ (𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑))) → ((if(𝑐𝑅𝑑, 𝑐, 𝑑) = 𝑎𝑐 = 𝑏) → ((𝑐 = 𝑎𝑑 = 𝑏) ∨ (𝑐 = 𝑏𝑑 = 𝑎))))
9580, 94sylbid 240 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 412 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1428 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 412 . . . . . . . . . . . . . . . . . . . . . . . . . 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 406 . . . . . . . . . . . . . . . . . . . . . . . 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 240 . . . . . . . . . . . . . . . . . . . . . . 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 412 . . . . . . . . . . . . . . . . . . . . . 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 412 . . . . . . . . . . . . . . . . . . . 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 2957 . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑏𝑅𝑎 ∧ (𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉))) → inf({𝑎, 𝑏}, 𝑉, 𝑅) = if(𝑎𝑅𝑏, 𝑎, 𝑏))
109 sotric 5637 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑅 Or 𝑉 ∧ (𝑏𝑉𝑎𝑉)) → (𝑏𝑅𝑎 ↔ ¬ (𝑏 = 𝑎𝑎𝑅𝑏)))
110109ancom2s 649 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉)) → (𝑏𝑅𝑎 ↔ ¬ (𝑏 = 𝑎𝑎𝑅𝑏)))
111 ioran 984 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (¬ (𝑏 = 𝑎𝑎𝑅𝑏) ↔ (¬ 𝑏 = 𝑎 ∧ ¬ 𝑎𝑅𝑏))
112 iffalse 4557 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 𝑎𝑅𝑏 → if(𝑎𝑅𝑏, 𝑎, 𝑏) = 𝑏)
113111, 112simplbiim 504 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (¬ (𝑏 = 𝑎𝑎𝑅𝑏) → if(𝑎𝑅𝑏, 𝑎, 𝑏) = 𝑏)
114110, 113biimtrdi 253 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉)) → (𝑏𝑅𝑎 → if(𝑎𝑅𝑏, 𝑎, 𝑏) = 𝑏))
115114impcom 407 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑏𝑅𝑎 ∧ (𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉))) → if(𝑎𝑅𝑏, 𝑎, 𝑏) = 𝑏)
116108, 115eqtrd 2780 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑏𝑅𝑎 ∧ (𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉))) → inf({𝑎, 𝑏}, 𝑉, 𝑅) = 𝑏)
117116eqeq2d 2751 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑏𝑅𝑎 ∧ (𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉))) → (inf({𝑐, 𝑑}, 𝑉, 𝑅) = inf({𝑎, 𝑏}, 𝑉, 𝑅) ↔ inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏))
118 iftrue 4554 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑏𝑅𝑎 → if(𝑏𝑅𝑎, 𝑎, 𝑏) = 𝑎)
11934, 118sylan9eqr 2802 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑏𝑅𝑎 ∧ (𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉))) → sup({𝑎, 𝑏}, 𝑉, 𝑅) = 𝑎)
120119eqeq2d 2751 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑏𝑅𝑎 ∧ (𝑅 Or 𝑉 ∧ (𝑎𝑉𝑏𝑉))) → (sup({𝑐, 𝑑}, 𝑉, 𝑅) = sup({𝑎, 𝑏}, 𝑉, 𝑅) ↔ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎))
121117, 120anbi12d 631 . . . . . . . . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . . . . . . . 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 2742 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑐𝑅𝑑 ∧ (𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑))) → (inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏𝑐 = 𝑏))
12466eqeq1d 2742 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑐𝑅𝑑 ∧ (𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑))) → (sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎𝑑 = 𝑎))
125123, 124anbi12d 631 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑐𝑅𝑑 ∧ (𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑))) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎) ↔ (𝑐 = 𝑏𝑑 = 𝑎)))
126125, 92biimtrdi 253 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 412 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2957 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑐 = 𝑑 → (𝑐𝑑 → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎) → ((𝑐 = 𝑎𝑑 = 𝑏) ∨ (𝑐 = 𝑏𝑑 = 𝑎)))))
129128adantld 490 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑐 = 𝑑 → (((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎) → ((𝑐 = 𝑎𝑑 = 𝑏) ∨ (𝑐 = 𝑏𝑑 = 𝑎)))))
130129adantld 490 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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, 87biimtrdi 253 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑)) → (𝑑𝑅𝑐 → if(𝑐𝑅𝑑, 𝑐, 𝑑) = 𝑑))
132131impcom 407 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑑𝑅𝑐 ∧ (𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑))) → if(𝑐𝑅𝑑, 𝑐, 𝑑) = 𝑑)
13375, 132eqtrd 2780 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑑𝑅𝑐 ∧ (𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑))) → inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑑)
134133eqeq1d 2742 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑑𝑅𝑐 ∧ (𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑))) → (inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏𝑑 = 𝑏))
13578eqeq1d 2742 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑑𝑅𝑐 ∧ (𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑))) → (sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎𝑐 = 𝑎))
136134, 135anbi12d 631 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑑𝑅𝑐 ∧ (𝑅 Or 𝑉 ∧ ((𝑐𝑉𝑑𝑉) ∧ 𝑐𝑑))) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎) ↔ (𝑑 = 𝑏𝑐 = 𝑎)))
13769ancoms 458 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑑 = 𝑏𝑐 = 𝑎) → ((𝑐 = 𝑎𝑑 = 𝑏) ∨ (𝑐 = 𝑏𝑑 = 𝑎)))
138136, 137biimtrdi 253 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 412 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1428 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 412 . . . . . . . . . . . . . . . . . . . . . . . . . 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 406 . . . . . . . . . . . . . . . . . . . . . . . 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 240 . . . . . . . . . . . . . . . . . . . . . . 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 412 . . . . . . . . . . . . . . . . . . . . . 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 412 . . . . . . . . . . . . . . . . . . . 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 1428 . . . . . . . . . . . . . . . . . . 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 490 . . . . . . . . . . . . . . . . 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 406 . . . . . . . . . . . . . . . 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 452 . . . . . . . . . . . . . . 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 490 . . . . . . . . . . . . . 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 406 . . . . . . . . . . . . 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 3492 . . . . . . . . . . . . . 14 𝑐 ∈ V
157 vex 3492 . . . . . . . . . . . . . 14 𝑑 ∈ V
158 vex 3492 . . . . . . . . . . . . . 14 𝑎 ∈ V
159 vex 3492 . . . . . . . . . . . . . 14 𝑏 ∈ V
160156, 157, 158, 159preq12b 4875 . . . . . . . . . . . . 13 ({𝑐, 𝑑} = {𝑎, 𝑏} ↔ ((𝑐 = 𝑎𝑑 = 𝑏) ∨ (𝑐 = 𝑏𝑑 = 𝑎)))
161155, 160imbitrrdi 252 . . . . . . . . . . . 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 240 . . . . . . . . . . 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 9545 . . . . . . . . . . . . . . . . . . . 20 (𝑍 = {𝑐, 𝑑} → inf(𝑍, 𝑉, 𝑅) = inf({𝑐, 𝑑}, 𝑉, 𝑅))
164 supeq1 9514 . . . . . . . . . . . . . . . . . . . 20 (𝑍 = {𝑐, 𝑑} → sup(𝑍, 𝑉, 𝑅) = sup({𝑐, 𝑑}, 𝑉, 𝑅))
165163, 164opeq12d 4905 . . . . . . . . . . . . . . . . . . 19 (𝑍 = {𝑐, 𝑑} → ⟨inf(𝑍, 𝑉, 𝑅), sup(𝑍, 𝑉, 𝑅)⟩ = ⟨inf({𝑐, 𝑑}, 𝑉, 𝑅), sup({𝑐, 𝑑}, 𝑉, 𝑅)⟩)
166 infeq1 9545 . . . . . . . . . . . . . . . . . . . 20 (𝑊 = {𝑎, 𝑏} → inf(𝑊, 𝑉, 𝑅) = inf({𝑎, 𝑏}, 𝑉, 𝑅))
167 supeq1 9514 . . . . . . . . . . . . . . . . . . . 20 (𝑊 = {𝑎, 𝑏} → sup(𝑊, 𝑉, 𝑅) = sup({𝑎, 𝑏}, 𝑉, 𝑅))
168166, 167opeq12d 4905 . . . . . . . . . . . . . . . . . . 19 (𝑊 = {𝑎, 𝑏} → ⟨inf(𝑊, 𝑉, 𝑅), sup(𝑊, 𝑉, 𝑅)⟩ = ⟨inf({𝑎, 𝑏}, 𝑉, 𝑅), sup({𝑎, 𝑏}, 𝑉, 𝑅)⟩)
169165, 168eqeqan12rd 2755 . . . . . . . . . . . . . . . . . 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 2757 . . . . . . . . . . . . . . . . . . 19 ((𝑍 = {𝑐, 𝑑} ∧ 𝑊 = {𝑎, 𝑏}) → (𝑍 = 𝑊 ↔ {𝑐, 𝑑} = {𝑎, 𝑏}))
171170ancoms 458 . . . . . . . . . . . . . . . . . 18 ((𝑊 = {𝑎, 𝑏} ∧ 𝑍 = {𝑐, 𝑑}) → (𝑍 = 𝑊 ↔ {𝑐, 𝑑} = {𝑎, 𝑏}))
172169, 171imbi12d 344 . . . . . . . . . . . . . . . . 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 412 . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . 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 480 . . . . . . . . . . . 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 407 . . . . . . . . . . 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 257 . . . . . . . . . 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 412 . . . . . . . . 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 3219 . . . . . . . 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 412 . . . . . . 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 3219 . . . . . 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, 184biimtrid 242 . . . 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 217 . . 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 1112 . 2 ((𝑅 Or 𝑉𝑊𝑃𝑍𝑃) → (⟨inf(𝑍, 𝑉, 𝑅), sup(𝑍, 𝑉, 𝑅)⟩ = ⟨inf(𝑊, 𝑉, 𝑅), sup(𝑊, 𝑉, 𝑅)⟩ → 𝑍 = 𝑊))
18816, 187sylbid 240 1 ((𝑅 Or 𝑉𝑊𝑃𝑍𝑃) → ((𝐹𝑍) = (𝐹𝑊) → 𝑍 = 𝑊))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 846  w3o 1086  w3a 1087   = wceq 1537  wcel 2108  wne 2946  wrex 3076  {crab 3443  Vcvv 3488  cin 3975  ifcif 4548  𝒫 cpw 4622  {cpr 4650  cop 4654   class class class wbr 5166  cmpt 5249   Or wor 5606   × cxp 5698  cfv 6573  supcsup 9509  infcinf 9510  2c2 12348  chash 14379
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-oadd 8526  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-sup 9511  df-inf 9512  df-dju 9970  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-n0 12554  df-z 12640  df-uz 12904  df-fz 13568  df-hash 14380
This theorem is referenced by:  prproropf1o  47381
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