Theorem prproropf1olem4 48510

Description: Lemma 4 for prproropf1o 48511. (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 = (𝑝 ∈ 𝑃 ↦ ⟨inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)⟩)

Assertion

Ref	Expression
prproropf1olem4	⊢ ((𝑅 Or 𝑉 ∧ 𝑊 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃) → ((𝐹‘𝑍) = (𝐹‘𝑊) → 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.

Step	Hyp	Ref	Expression
1		prproropf1o.f	. . . 4 ⊢ 𝐹 = (𝑝 ∈ 𝑃 ↦ ⟨inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)⟩)
2		infeq1 9447	. . . . 5 ⊢ (𝑝 = 𝑍 → inf(𝑝, 𝑉, 𝑅) = inf(𝑍, 𝑉, 𝑅))
3		supeq1 9415	. . . . 5 ⊢ (𝑝 = 𝑍 → sup(𝑝, 𝑉, 𝑅) = sup(𝑍, 𝑉, 𝑅))
4	2, 3	opeq12d 4840	. . . 4 ⊢ (𝑝 = 𝑍 → ⟨inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)⟩ = ⟨inf(𝑍, 𝑉, 𝑅), sup(𝑍, 𝑉, 𝑅)⟩)
5		simp3 1156	. . . 4 ⊢ ((𝑅 Or 𝑉 ∧ 𝑊 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃) → 𝑍 ∈ 𝑃)
6		opex 5431	. . . . 5 ⊢ ⟨inf(𝑍, 𝑉, 𝑅), sup(𝑍, 𝑉, 𝑅)⟩ ∈ V
7	6	a1i 11	. . . 4 ⊢ ((𝑅 Or 𝑉 ∧ 𝑊 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃) → ⟨inf(𝑍, 𝑉, 𝑅), sup(𝑍, 𝑉, 𝑅)⟩ ∈ V)
8	1, 4, 5, 7	fvmptd3 7005	. . 3 ⊢ ((𝑅 Or 𝑉 ∧ 𝑊 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃) → (𝐹‘𝑍) = ⟨inf(𝑍, 𝑉, 𝑅), sup(𝑍, 𝑉, 𝑅)⟩)
9		infeq1 9447	. . . . 5 ⊢ (𝑝 = 𝑊 → inf(𝑝, 𝑉, 𝑅) = inf(𝑊, 𝑉, 𝑅))
10		supeq1 9415	. . . . 5 ⊢ (𝑝 = 𝑊 → sup(𝑝, 𝑉, 𝑅) = sup(𝑊, 𝑉, 𝑅))
11	9, 10	opeq12d 4840	. . . 4 ⊢ (𝑝 = 𝑊 → ⟨inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)⟩ = ⟨inf(𝑊, 𝑉, 𝑅), sup(𝑊, 𝑉, 𝑅)⟩)
12		simp2 1155	. . . 4 ⊢ ((𝑅 Or 𝑉 ∧ 𝑊 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃) → 𝑊 ∈ 𝑃)
13		opex 5431	. . . . 5 ⊢ ⟨inf(𝑊, 𝑉, 𝑅), sup(𝑊, 𝑉, 𝑅)⟩ ∈ V
14	13	a1i 11	. . . 4 ⊢ ((𝑅 Or 𝑉 ∧ 𝑊 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃) → ⟨inf(𝑊, 𝑉, 𝑅), sup(𝑊, 𝑉, 𝑅)⟩ ∈ V)
15	1, 11, 12, 14	fvmptd3 7005	. . 3 ⊢ ((𝑅 Or 𝑉 ∧ 𝑊 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃) → (𝐹‘𝑊) = ⟨inf(𝑊, 𝑉, 𝑅), sup(𝑊, 𝑉, 𝑅)⟩)
16	8, 15	eqeq12d 2776	. 2 ⊢ ((𝑅 Or 𝑉 ∧ 𝑊 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃) → ((𝐹‘𝑍) = (𝐹‘𝑊) ↔ ⟨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}
18	17	prpair 48505	. . . 4 ⊢ (𝑍 ∈ 𝑃 ↔ ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (𝑍 = {𝑐, 𝑑} ∧ 𝑐 ≠ 𝑑))
19	17	prpair 48505	. . . . 5 ⊢ (𝑊 ∈ 𝑃 ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑊 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏))
20		id 23	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 Or 𝑉 → 𝑅 Or 𝑉)
21	20	infexd 9454	. . . . . . . . . . . . . . 15 ⊢ (𝑅 Or 𝑉 → inf({𝑐, 𝑑}, 𝑉, 𝑅) ∈ V)
22	20	supexd 9423	. . . . . . . . . . . . . . 15 ⊢ (𝑅 Or 𝑉 → sup({𝑐, 𝑑}, 𝑉, 𝑅) ∈ V)
23	21, 22	jca 521	. . . . . . . . . . . . . 14 ⊢ (𝑅 Or 𝑉 → (inf({𝑐, 𝑑}, 𝑉, 𝑅) ∈ V ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) ∈ V))
24	23	ad4antr 745	. . . . . . . . . . . . 13 ⊢ (((((𝑅 Or 𝑉 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑊 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) ∧ (𝑍 = {𝑐, 𝑑} ∧ 𝑐 ≠ 𝑑)) → (inf({𝑐, 𝑑}, 𝑉, 𝑅) ∈ V ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) ∈ V))
25		opthg 5445	. . . . . . . . . . . . 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))))
26	24, 25	syl 18	. . . . . . . . . . . 12 ⊢ (((((𝑅 Or 𝑉 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑊 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) ∧ (𝑍 = {𝑐, 𝑑} ∧ 𝑐 ≠ 𝑑)) → (⟨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 5582	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑅 Or 𝑉 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝑎𝑅𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑅𝑎))
28		infpr 9475	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑅 Or 𝑉 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → inf({𝑎, 𝑏}, 𝑉, 𝑅) = if(𝑎𝑅𝑏, 𝑎, 𝑏))
29	28	3expb 1138	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑅 Or 𝑉 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → inf({𝑎, 𝑏}, 𝑉, 𝑅) = if(𝑎𝑅𝑏, 𝑎, 𝑏))
30		iftrue 4487	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑎𝑅𝑏 → if(𝑎𝑅𝑏, 𝑎, 𝑏) = 𝑎)
31	29, 30	sylan9eqr 2817	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑎𝑅𝑏 ∧ (𝑅 Or 𝑉 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉))) → inf({𝑎, 𝑏}, 𝑉, 𝑅) = 𝑎)
32	31	eqeq2d 2771	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑎𝑅𝑏 ∧ (𝑅 Or 𝑉 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉))) → (inf({𝑐, 𝑑}, 𝑉, 𝑅) = inf({𝑎, 𝑏}, 𝑉, 𝑅) ↔ inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎))
33		suppr 9442	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑅 Or 𝑉 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → sup({𝑎, 𝑏}, 𝑉, 𝑅) = if(𝑏𝑅𝑎, 𝑎, 𝑏))
34	33	3expb 1138	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑅 Or 𝑉 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → sup({𝑎, 𝑏}, 𝑉, 𝑅) = if(𝑏𝑅𝑎, 𝑎, 𝑏))
35	34	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑎𝑅𝑏 ∧ (𝑅 Or 𝑉 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉))) → sup({𝑎, 𝑏}, 𝑉, 𝑅) = if(𝑏𝑅𝑎, 𝑎, 𝑏))
36		sotric 5585	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑅 Or 𝑉 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝑎𝑅𝑏 ↔ ¬ (𝑎 = 𝑏 ∨ 𝑏𝑅𝑎)))
37		ioran 999	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (¬ (𝑎 = 𝑏 ∨ 𝑏𝑅𝑎) ↔ (¬ 𝑎 = 𝑏 ∧ ¬ 𝑏𝑅𝑎))
38		iffalse 4490	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (¬ 𝑏𝑅𝑎 → if(𝑏𝑅𝑎, 𝑎, 𝑏) = 𝑏)
39	37, 38	simplbiim 514	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (¬ (𝑎 = 𝑏 ∨ 𝑏𝑅𝑎) → if(𝑏𝑅𝑎, 𝑎, 𝑏) = 𝑏)
40	36, 39	biimtrdi 256	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑅 Or 𝑉 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝑎𝑅𝑏 → if(𝑏𝑅𝑎, 𝑎, 𝑏) = 𝑏))
41	40	impcom 413	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑎𝑅𝑏 ∧ (𝑅 Or 𝑉 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉))) → if(𝑏𝑅𝑎, 𝑎, 𝑏) = 𝑏)
42	35, 41	eqtrd 2795	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑎𝑅𝑏 ∧ (𝑅 Or 𝑉 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉))) → sup({𝑎, 𝑏}, 𝑉, 𝑅) = 𝑏)
43	42	eqeq2d 2771	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑎𝑅𝑏 ∧ (𝑅 Or 𝑉 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉))) → (sup({𝑐, 𝑑}, 𝑉, 𝑅) = sup({𝑎, 𝑏}, 𝑉, 𝑅) ↔ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏))
44	32, 43	anbi12d 644	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑎𝑅𝑏 ∧ (𝑅 Or 𝑉 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉))) → ((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)))
45	44	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑎𝑅𝑏 ∧ (𝑅 Or 𝑉 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉))) ∧ ((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑)) → ((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 5582	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑅 Or 𝑉 ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → (𝑐𝑅𝑑 ∨ 𝑐 = 𝑑 ∨ 𝑑𝑅𝑐))
47	46	adantrr 730	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑅 Or 𝑉 ∧ ((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑)) → (𝑐𝑅𝑑 ∨ 𝑐 = 𝑑 ∨ 𝑑𝑅𝑐))
48		simpl 488	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑅 Or 𝑉 ∧ ((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑)) → 𝑅 Or 𝑉)
49		simprll 791	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑅 Or 𝑉 ∧ ((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑)) → 𝑐 ∈ 𝑉)
50		simprlr 792	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑅 Or 𝑉 ∧ ((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑)) → 𝑑 ∈ 𝑉)
51		infpr 9475	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑅 Or 𝑉 ∧ 𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) → inf({𝑐, 𝑑}, 𝑉, 𝑅) = if(𝑐𝑅𝑑, 𝑐, 𝑑))
52	48, 49, 50, 51	syl3anc 1398	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑅 Or 𝑉 ∧ ((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑)) → inf({𝑐, 𝑑}, 𝑉, 𝑅) = if(𝑐𝑅𝑑, 𝑐, 𝑑))
53		iftrue 4487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑐𝑅𝑑 → if(𝑐𝑅𝑑, 𝑐, 𝑑) = 𝑐)
54	52, 53	sylan9eqr 2817	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑐𝑅𝑑 ∧ (𝑅 Or 𝑉 ∧ ((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑))) → inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑐)
55	54	eqeq1d 2762	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑐𝑅𝑑 ∧ (𝑅 Or 𝑉 ∧ ((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑))) → (inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎 ↔ 𝑐 = 𝑎))
56		suppr 9442	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑅 Or 𝑉 ∧ 𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) → sup({𝑐, 𝑑}, 𝑉, 𝑅) = if(𝑑𝑅𝑐, 𝑐, 𝑑))
57	48, 49, 50, 56	syl3anc 1398	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑅 Or 𝑉 ∧ ((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑)) → sup({𝑐, 𝑑}, 𝑉, 𝑅) = if(𝑑𝑅𝑐, 𝑐, 𝑑))
58	57	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑐𝑅𝑑 ∧ (𝑅 Or 𝑉 ∧ ((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑))) → sup({𝑐, 𝑑}, 𝑉, 𝑅) = if(𝑑𝑅𝑐, 𝑐, 𝑑))
59		sotric 5585	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑅 Or 𝑉 ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → (𝑐𝑅𝑑 ↔ ¬ (𝑐 = 𝑑 ∨ 𝑑𝑅𝑐)))
60	59	adantrr 730	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑅 Or 𝑉 ∧ ((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑)) → (𝑐𝑅𝑑 ↔ ¬ (𝑐 = 𝑑 ∨ 𝑑𝑅𝑐)))
61		ioran 999	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (¬ (𝑐 = 𝑑 ∨ 𝑑𝑅𝑐) ↔ (¬ 𝑐 = 𝑑 ∧ ¬ 𝑑𝑅𝑐))
62		iffalse 4490	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (¬ 𝑑𝑅𝑐 → if(𝑑𝑅𝑐, 𝑐, 𝑑) = 𝑑)
63	61, 62	simplbiim 514	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (¬ (𝑐 = 𝑑 ∨ 𝑑𝑅𝑐) → if(𝑑𝑅𝑐, 𝑐, 𝑑) = 𝑑)
64	60, 63	biimtrdi 256	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑅 Or 𝑉 ∧ ((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑)) → (𝑐𝑅𝑑 → if(𝑑𝑅𝑐, 𝑐, 𝑑) = 𝑑))
65	64	impcom 413	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑐𝑅𝑑 ∧ (𝑅 Or 𝑉 ∧ ((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑))) → if(𝑑𝑅𝑐, 𝑐, 𝑑) = 𝑑)
66	58, 65	eqtrd 2795	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑐𝑅𝑑 ∧ (𝑅 Or 𝑉 ∧ ((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑))) → sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑑)
67	66	eqeq1d 2762	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑐𝑅𝑑 ∧ (𝑅 Or 𝑉 ∧ ((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑))) → (sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏 ↔ 𝑑 = 𝑏))
68	55, 67	anbi12d 644	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑐𝑅𝑑 ∧ (𝑅 Or 𝑉 ∧ ((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑))) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏) ↔ (𝑐 = 𝑎 ∧ 𝑑 = 𝑏)))
69		orc 881	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑐 = 𝑎 ∧ 𝑑 = 𝑏) → ((𝑐 = 𝑎 ∧ 𝑑 = 𝑏) ∨ (𝑐 = 𝑏 ∧ 𝑑 = 𝑎)))
70	68, 69	biimtrdi 256	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑐𝑅𝑑 ∧ (𝑅 Or 𝑉 ∧ ((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑))) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏) → ((𝑐 = 𝑎 ∧ 𝑑 = 𝑏) ∨ (𝑐 = 𝑏 ∧ 𝑑 = 𝑎))))
71	70	ex 418	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑐𝑅𝑑 → ((𝑅 Or 𝑉 ∧ ((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑)) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏) → ((𝑐 = 𝑎 ∧ 𝑑 = 𝑏) ∨ (𝑐 = 𝑏 ∧ 𝑑 = 𝑎)))))
72		eqneqall 2966	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑐 = 𝑑 → (𝑐 ≠ 𝑑 → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏) → ((𝑐 = 𝑎 ∧ 𝑑 = 𝑏) ∨ (𝑐 = 𝑏 ∧ 𝑑 = 𝑎)))))
73	72	adantld 496	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑐 = 𝑑 → (((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏) → ((𝑐 = 𝑎 ∧ 𝑑 = 𝑏) ∨ (𝑐 = 𝑏 ∧ 𝑑 = 𝑎)))))
74	73	adantld 496	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑐 = 𝑑 → ((𝑅 Or 𝑉 ∧ ((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑)) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏) → ((𝑐 = 𝑎 ∧ 𝑑 = 𝑏) ∨ (𝑐 = 𝑏 ∧ 𝑑 = 𝑎)))))
75	52	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑑𝑅𝑐 ∧ (𝑅 Or 𝑉 ∧ ((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑))) → inf({𝑐, 𝑑}, 𝑉, 𝑅) = if(𝑐𝑅𝑑, 𝑐, 𝑑))
76	75	eqeq1d 2762	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑑𝑅𝑐 ∧ (𝑅 Or 𝑉 ∧ ((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑))) → (inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎 ↔ if(𝑐𝑅𝑑, 𝑐, 𝑑) = 𝑎))
77		iftrue 4487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑑𝑅𝑐 → if(𝑑𝑅𝑐, 𝑐, 𝑑) = 𝑐)
78	57, 77	sylan9eqr 2817	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑑𝑅𝑐 ∧ (𝑅 Or 𝑉 ∧ ((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑))) → sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑐)
79	78	eqeq1d 2762	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑑𝑅𝑐 ∧ (𝑅 Or 𝑉 ∧ ((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑))) → (sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏 ↔ 𝑐 = 𝑏))
80	76, 79	anbi12d 644	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑑𝑅𝑐 ∧ (𝑅 Or 𝑉 ∧ ((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑))) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏) ↔ (if(𝑐𝑅𝑑, 𝑐, 𝑑) = 𝑎 ∧ 𝑐 = 𝑏)))
81		simpl 488	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑) → (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))
82	81	ancomd 467	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑) → (𝑑 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉))
83		sotric 5585	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑅 Or 𝑉 ∧ (𝑑 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (𝑑𝑅𝑐 ↔ ¬ (𝑑 = 𝑐 ∨ 𝑐𝑅𝑑)))
84	82, 83	sylan2 605	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑅 Or 𝑉 ∧ ((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑)) → (𝑑𝑅𝑐 ↔ ¬ (𝑑 = 𝑐 ∨ 𝑐𝑅𝑑)))
85		ioran 999	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (¬ (𝑑 = 𝑐 ∨ 𝑐𝑅𝑑) ↔ (¬ 𝑑 = 𝑐 ∧ ¬ 𝑐𝑅𝑑))
86		iffalse 4490	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (¬ 𝑐𝑅𝑑 → if(𝑐𝑅𝑑, 𝑐, 𝑑) = 𝑑)
87	85, 86	simplbiim 514	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (¬ (𝑑 = 𝑐 ∨ 𝑐𝑅𝑑) → if(𝑐𝑅𝑑, 𝑐, 𝑑) = 𝑑)
88	87	eqeq1d 2762	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (¬ (𝑑 = 𝑐 ∨ 𝑐𝑅𝑑) → (if(𝑐𝑅𝑑, 𝑐, 𝑑) = 𝑎 ↔ 𝑑 = 𝑎))
89	84, 88	biimtrdi 256	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑅 Or 𝑉 ∧ ((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑)) → (𝑑𝑅𝑐 → (if(𝑐𝑅𝑑, 𝑐, 𝑑) = 𝑎 ↔ 𝑑 = 𝑎)))
90	89	impcom 413	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑑𝑅𝑐 ∧ (𝑅 Or 𝑉 ∧ ((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑))) → (if(𝑐𝑅𝑑, 𝑐, 𝑑) = 𝑎 ↔ 𝑑 = 𝑎))
91	90	anbi1d 643	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑑𝑅𝑐 ∧ (𝑅 Or 𝑉 ∧ ((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑))) → ((if(𝑐𝑅𝑑, 𝑐, 𝑑) = 𝑎 ∧ 𝑐 = 𝑏) ↔ (𝑑 = 𝑎 ∧ 𝑐 = 𝑏)))
92		olc 882	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑐 = 𝑏 ∧ 𝑑 = 𝑎) → ((𝑐 = 𝑎 ∧ 𝑑 = 𝑏) ∨ (𝑐 = 𝑏 ∧ 𝑑 = 𝑎)))
93	92	ancoms 464	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑑 = 𝑎 ∧ 𝑐 = 𝑏) → ((𝑐 = 𝑎 ∧ 𝑑 = 𝑏) ∨ (𝑐 = 𝑏 ∧ 𝑑 = 𝑎)))
94	91, 93	biimtrdi 256	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑑𝑅𝑐 ∧ (𝑅 Or 𝑉 ∧ ((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑))) → ((if(𝑐𝑅𝑑, 𝑐, 𝑑) = 𝑎 ∧ 𝑐 = 𝑏) → ((𝑐 = 𝑎 ∧ 𝑑 = 𝑏) ∨ (𝑐 = 𝑏 ∧ 𝑑 = 𝑎))))
95	80, 94	sylbid 243	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑑𝑅𝑐 ∧ (𝑅 Or 𝑉 ∧ ((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑))) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏) → ((𝑐 = 𝑎 ∧ 𝑑 = 𝑏) ∨ (𝑐 = 𝑏 ∧ 𝑑 = 𝑎))))
96	95	ex 418	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑑𝑅𝑐 → ((𝑅 Or 𝑉 ∧ ((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑)) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏) → ((𝑐 = 𝑎 ∧ 𝑑 = 𝑏) ∨ (𝑐 = 𝑏 ∧ 𝑑 = 𝑎)))))
97	71, 74, 96	3jaoi 1454	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑐𝑅𝑑 ∨ 𝑐 = 𝑑 ∨ 𝑑𝑅𝑐) → ((𝑅 Or 𝑉 ∧ ((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑)) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏) → ((𝑐 = 𝑎 ∧ 𝑑 = 𝑏) ∨ (𝑐 = 𝑏 ∧ 𝑑 = 𝑎)))))
98	47, 97	mpcom 39	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑅 Or 𝑉 ∧ ((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑)) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏) → ((𝑐 = 𝑎 ∧ 𝑑 = 𝑏) ∨ (𝑐 = 𝑏 ∧ 𝑑 = 𝑎))))
99	98	ex 418	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑅 Or 𝑉 → (((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏) → ((𝑐 = 𝑎 ∧ 𝑑 = 𝑏) ∨ (𝑐 = 𝑏 ∧ 𝑑 = 𝑎)))))
100	99	ad2antrl 741	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑎𝑅𝑏 ∧ (𝑅 Or 𝑉 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉))) → (((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏) → ((𝑐 = 𝑎 ∧ 𝑑 = 𝑏) ∨ (𝑐 = 𝑏 ∧ 𝑑 = 𝑎)))))
101	100	imp 412	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑎𝑅𝑏 ∧ (𝑅 Or 𝑉 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉))) ∧ ((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑)) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏) → ((𝑐 = 𝑎 ∧ 𝑑 = 𝑏) ∨ (𝑐 = 𝑏 ∧ 𝑑 = 𝑎))))
102	45, 101	sylbid 243	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑎𝑅𝑏 ∧ (𝑅 Or 𝑉 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉))) ∧ ((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑)) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = inf({𝑎, 𝑏}, 𝑉, 𝑅) ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = sup({𝑎, 𝑏}, 𝑉, 𝑅)) → ((𝑐 = 𝑎 ∧ 𝑑 = 𝑏) ∨ (𝑐 = 𝑏 ∧ 𝑑 = 𝑎))))
103	102	ex 418	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑎𝑅𝑏 ∧ (𝑅 Or 𝑉 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉))) → (((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = inf({𝑎, 𝑏}, 𝑉, 𝑅) ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = sup({𝑎, 𝑏}, 𝑉, 𝑅)) → ((𝑐 = 𝑎 ∧ 𝑑 = 𝑏) ∨ (𝑐 = 𝑏 ∧ 𝑑 = 𝑎)))))
104	103	a1d 26	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑎𝑅𝑏 ∧ (𝑅 Or 𝑉 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉))) → (𝑎 ≠ 𝑏 → (((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = inf({𝑎, 𝑏}, 𝑉, 𝑅) ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = sup({𝑎, 𝑏}, 𝑉, 𝑅)) → ((𝑐 = 𝑎 ∧ 𝑑 = 𝑏) ∨ (𝑐 = 𝑏 ∧ 𝑑 = 𝑎))))))
105	104	ex 418	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎𝑅𝑏 → ((𝑅 Or 𝑉 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝑎 ≠ 𝑏 → (((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = inf({𝑎, 𝑏}, 𝑉, 𝑅) ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = sup({𝑎, 𝑏}, 𝑉, 𝑅)) → ((𝑐 = 𝑎 ∧ 𝑑 = 𝑏) ∨ (𝑐 = 𝑏 ∧ 𝑑 = 𝑎)))))))
106		eqneqall 2966	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 = 𝑏 → (𝑎 ≠ 𝑏 → (((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = inf({𝑎, 𝑏}, 𝑉, 𝑅) ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = sup({𝑎, 𝑏}, 𝑉, 𝑅)) → ((𝑐 = 𝑎 ∧ 𝑑 = 𝑏) ∨ (𝑐 = 𝑏 ∧ 𝑑 = 𝑎))))))
107	106	a1d 26	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝑏 → ((𝑅 Or 𝑉 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝑎 ≠ 𝑏 → (((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = inf({𝑎, 𝑏}, 𝑉, 𝑅) ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = sup({𝑎, 𝑏}, 𝑉, 𝑅)) → ((𝑐 = 𝑎 ∧ 𝑑 = 𝑏) ∨ (𝑐 = 𝑏 ∧ 𝑑 = 𝑎)))))))
108	29	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑏𝑅𝑎 ∧ (𝑅 Or 𝑉 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉))) → inf({𝑎, 𝑏}, 𝑉, 𝑅) = if(𝑎𝑅𝑏, 𝑎, 𝑏))
109		sotric 5585	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑅 Or 𝑉 ∧ (𝑏 ∈ 𝑉 ∧ 𝑎 ∈ 𝑉)) → (𝑏𝑅𝑎 ↔ ¬ (𝑏 = 𝑎 ∨ 𝑎𝑅𝑏)))
110	109	ancom2s 663	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑅 Or 𝑉 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝑏𝑅𝑎 ↔ ¬ (𝑏 = 𝑎 ∨ 𝑎𝑅𝑏)))
111		ioran 999	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (¬ (𝑏 = 𝑎 ∨ 𝑎𝑅𝑏) ↔ (¬ 𝑏 = 𝑎 ∧ ¬ 𝑎𝑅𝑏))
112		iffalse 4490	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (¬ 𝑎𝑅𝑏 → if(𝑎𝑅𝑏, 𝑎, 𝑏) = 𝑏)
113	111, 112	simplbiim 514	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (¬ (𝑏 = 𝑎 ∨ 𝑎𝑅𝑏) → if(𝑎𝑅𝑏, 𝑎, 𝑏) = 𝑏)
114	110, 113	biimtrdi 256	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑅 Or 𝑉 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝑏𝑅𝑎 → if(𝑎𝑅𝑏, 𝑎, 𝑏) = 𝑏))
115	114	impcom 413	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑏𝑅𝑎 ∧ (𝑅 Or 𝑉 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉))) → if(𝑎𝑅𝑏, 𝑎, 𝑏) = 𝑏)
116	108, 115	eqtrd 2795	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑏𝑅𝑎 ∧ (𝑅 Or 𝑉 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉))) → inf({𝑎, 𝑏}, 𝑉, 𝑅) = 𝑏)
117	116	eqeq2d 2771	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑏𝑅𝑎 ∧ (𝑅 Or 𝑉 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉))) → (inf({𝑐, 𝑑}, 𝑉, 𝑅) = inf({𝑎, 𝑏}, 𝑉, 𝑅) ↔ inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏))
118		iftrue 4487	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑏𝑅𝑎 → if(𝑏𝑅𝑎, 𝑎, 𝑏) = 𝑎)
119	34, 118	sylan9eqr 2817	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑏𝑅𝑎 ∧ (𝑅 Or 𝑉 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉))) → sup({𝑎, 𝑏}, 𝑉, 𝑅) = 𝑎)
120	119	eqeq2d 2771	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑏𝑅𝑎 ∧ (𝑅 Or 𝑉 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉))) → (sup({𝑐, 𝑑}, 𝑉, 𝑅) = sup({𝑎, 𝑏}, 𝑉, 𝑅) ↔ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎))
121	117, 120	anbi12d 644	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑏𝑅𝑎 ∧ (𝑅 Or 𝑉 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉))) → ((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)))
122	121	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑏𝑅𝑎 ∧ (𝑅 Or 𝑉 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉))) ∧ ((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑)) → ((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)))
123	54	eqeq1d 2762	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑐𝑅𝑑 ∧ (𝑅 Or 𝑉 ∧ ((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑))) → (inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏 ↔ 𝑐 = 𝑏))
124	66	eqeq1d 2762	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑐𝑅𝑑 ∧ (𝑅 Or 𝑉 ∧ ((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑))) → (sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎 ↔ 𝑑 = 𝑎))
125	123, 124	anbi12d 644	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑐𝑅𝑑 ∧ (𝑅 Or 𝑉 ∧ ((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑))) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎) ↔ (𝑐 = 𝑏 ∧ 𝑑 = 𝑎)))
126	125, 92	biimtrdi 256	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑐𝑅𝑑 ∧ (𝑅 Or 𝑉 ∧ ((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑))) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎) → ((𝑐 = 𝑎 ∧ 𝑑 = 𝑏) ∨ (𝑐 = 𝑏 ∧ 𝑑 = 𝑎))))
127	126	ex 418	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑐𝑅𝑑 → ((𝑅 Or 𝑉 ∧ ((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑)) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎) → ((𝑐 = 𝑎 ∧ 𝑑 = 𝑏) ∨ (𝑐 = 𝑏 ∧ 𝑑 = 𝑎)))))
128		eqneqall 2966	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑐 = 𝑑 → (𝑐 ≠ 𝑑 → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎) → ((𝑐 = 𝑎 ∧ 𝑑 = 𝑏) ∨ (𝑐 = 𝑏 ∧ 𝑑 = 𝑎)))))
129	128	adantld 496	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑐 = 𝑑 → (((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎) → ((𝑐 = 𝑎 ∧ 𝑑 = 𝑏) ∨ (𝑐 = 𝑏 ∧ 𝑑 = 𝑎)))))
130	129	adantld 496	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑐 = 𝑑 → ((𝑅 Or 𝑉 ∧ ((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑)) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎) → ((𝑐 = 𝑎 ∧ 𝑑 = 𝑏) ∨ (𝑐 = 𝑏 ∧ 𝑑 = 𝑎)))))
131	84, 87	biimtrdi 256	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑅 Or 𝑉 ∧ ((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑)) → (𝑑𝑅𝑐 → if(𝑐𝑅𝑑, 𝑐, 𝑑) = 𝑑))
132	131	impcom 413	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑑𝑅𝑐 ∧ (𝑅 Or 𝑉 ∧ ((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑))) → if(𝑐𝑅𝑑, 𝑐, 𝑑) = 𝑑)
133	75, 132	eqtrd 2795	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑑𝑅𝑐 ∧ (𝑅 Or 𝑉 ∧ ((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑))) → inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑑)
134	133	eqeq1d 2762	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑑𝑅𝑐 ∧ (𝑅 Or 𝑉 ∧ ((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑))) → (inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏 ↔ 𝑑 = 𝑏))
135	78	eqeq1d 2762	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑑𝑅𝑐 ∧ (𝑅 Or 𝑉 ∧ ((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑))) → (sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎 ↔ 𝑐 = 𝑎))
136	134, 135	anbi12d 644	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑑𝑅𝑐 ∧ (𝑅 Or 𝑉 ∧ ((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑))) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎) ↔ (𝑑 = 𝑏 ∧ 𝑐 = 𝑎)))
137	69	ancoms 464	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑑 = 𝑏 ∧ 𝑐 = 𝑎) → ((𝑐 = 𝑎 ∧ 𝑑 = 𝑏) ∨ (𝑐 = 𝑏 ∧ 𝑑 = 𝑎)))
138	136, 137	biimtrdi 256	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑑𝑅𝑐 ∧ (𝑅 Or 𝑉 ∧ ((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑))) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎) → ((𝑐 = 𝑎 ∧ 𝑑 = 𝑏) ∨ (𝑐 = 𝑏 ∧ 𝑑 = 𝑎))))
139	138	ex 418	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑑𝑅𝑐 → ((𝑅 Or 𝑉 ∧ ((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑)) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎) → ((𝑐 = 𝑎 ∧ 𝑑 = 𝑏) ∨ (𝑐 = 𝑏 ∧ 𝑑 = 𝑎)))))
140	127, 130, 139	3jaoi 1454	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑐𝑅𝑑 ∨ 𝑐 = 𝑑 ∨ 𝑑𝑅𝑐) → ((𝑅 Or 𝑉 ∧ ((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑)) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎) → ((𝑐 = 𝑎 ∧ 𝑑 = 𝑏) ∨ (𝑐 = 𝑏 ∧ 𝑑 = 𝑎)))))
141	47, 140	mpcom 39	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑅 Or 𝑉 ∧ ((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑)) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎) → ((𝑐 = 𝑎 ∧ 𝑑 = 𝑏) ∨ (𝑐 = 𝑏 ∧ 𝑑 = 𝑎))))
142	141	ex 418	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑅 Or 𝑉 → (((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎) → ((𝑐 = 𝑎 ∧ 𝑑 = 𝑏) ∨ (𝑐 = 𝑏 ∧ 𝑑 = 𝑎)))))
143	142	ad2antrl 741	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑏𝑅𝑎 ∧ (𝑅 Or 𝑉 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉))) → (((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎) → ((𝑐 = 𝑎 ∧ 𝑑 = 𝑏) ∨ (𝑐 = 𝑏 ∧ 𝑑 = 𝑎)))))
144	143	imp 412	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑏𝑅𝑎 ∧ (𝑅 Or 𝑉 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉))) ∧ ((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑)) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑏 ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = 𝑎) → ((𝑐 = 𝑎 ∧ 𝑑 = 𝑏) ∨ (𝑐 = 𝑏 ∧ 𝑑 = 𝑎))))
145	122, 144	sylbid 243	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑏𝑅𝑎 ∧ (𝑅 Or 𝑉 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉))) ∧ ((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑)) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = inf({𝑎, 𝑏}, 𝑉, 𝑅) ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = sup({𝑎, 𝑏}, 𝑉, 𝑅)) → ((𝑐 = 𝑎 ∧ 𝑑 = 𝑏) ∨ (𝑐 = 𝑏 ∧ 𝑑 = 𝑎))))
146	145	ex 418	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑏𝑅𝑎 ∧ (𝑅 Or 𝑉 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉))) → (((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = inf({𝑎, 𝑏}, 𝑉, 𝑅) ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = sup({𝑎, 𝑏}, 𝑉, 𝑅)) → ((𝑐 = 𝑎 ∧ 𝑑 = 𝑏) ∨ (𝑐 = 𝑏 ∧ 𝑑 = 𝑎)))))
147	146	a1d 26	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑏𝑅𝑎 ∧ (𝑅 Or 𝑉 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉))) → (𝑎 ≠ 𝑏 → (((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = inf({𝑎, 𝑏}, 𝑉, 𝑅) ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = sup({𝑎, 𝑏}, 𝑉, 𝑅)) → ((𝑐 = 𝑎 ∧ 𝑑 = 𝑏) ∨ (𝑐 = 𝑏 ∧ 𝑑 = 𝑎))))))
148	147	ex 418	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑏𝑅𝑎 → ((𝑅 Or 𝑉 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝑎 ≠ 𝑏 → (((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = inf({𝑎, 𝑏}, 𝑉, 𝑅) ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = sup({𝑎, 𝑏}, 𝑉, 𝑅)) → ((𝑐 = 𝑎 ∧ 𝑑 = 𝑏) ∨ (𝑐 = 𝑏 ∧ 𝑑 = 𝑎)))))))
149	105, 107, 148	3jaoi 1454	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎𝑅𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑅𝑎) → ((𝑅 Or 𝑉 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝑎 ≠ 𝑏 → (((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = inf({𝑎, 𝑏}, 𝑉, 𝑅) ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = sup({𝑎, 𝑏}, 𝑉, 𝑅)) → ((𝑐 = 𝑎 ∧ 𝑑 = 𝑏) ∨ (𝑐 = 𝑏 ∧ 𝑑 = 𝑎)))))))
150	27, 149	mpcom 39	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑅 Or 𝑉 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝑎 ≠ 𝑏 → (((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = inf({𝑎, 𝑏}, 𝑉, 𝑅) ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = sup({𝑎, 𝑏}, 𝑉, 𝑅)) → ((𝑐 = 𝑎 ∧ 𝑑 = 𝑏) ∨ (𝑐 = 𝑏 ∧ 𝑑 = 𝑎))))))
151	150	adantld 496	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 Or 𝑉 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((𝑊 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏) → (((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = inf({𝑎, 𝑏}, 𝑉, 𝑅) ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = sup({𝑎, 𝑏}, 𝑉, 𝑅)) → ((𝑐 = 𝑎 ∧ 𝑑 = 𝑏) ∨ (𝑐 = 𝑏 ∧ 𝑑 = 𝑎))))))
152	151	imp 412	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 Or 𝑉 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑊 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)) → (((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ≠ 𝑑) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = inf({𝑎, 𝑏}, 𝑉, 𝑅) ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = sup({𝑎, 𝑏}, 𝑉, 𝑅)) → ((𝑐 = 𝑎 ∧ 𝑑 = 𝑏) ∨ (𝑐 = 𝑏 ∧ 𝑑 = 𝑎)))))
153	152	expdimp 458	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 Or 𝑉 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑊 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → (𝑐 ≠ 𝑑 → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = inf({𝑎, 𝑏}, 𝑉, 𝑅) ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = sup({𝑎, 𝑏}, 𝑉, 𝑅)) → ((𝑐 = 𝑎 ∧ 𝑑 = 𝑏) ∨ (𝑐 = 𝑏 ∧ 𝑑 = 𝑎)))))
154	153	adantld 496	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 Or 𝑉 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑊 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → ((𝑍 = {𝑐, 𝑑} ∧ 𝑐 ≠ 𝑑) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = inf({𝑎, 𝑏}, 𝑉, 𝑅) ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = sup({𝑎, 𝑏}, 𝑉, 𝑅)) → ((𝑐 = 𝑎 ∧ 𝑑 = 𝑏) ∨ (𝑐 = 𝑏 ∧ 𝑑 = 𝑎)))))
155	154	imp 412	. . . . . . . . . . . . 13 ⊢ (((((𝑅 Or 𝑉 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑊 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) ∧ (𝑍 = {𝑐, 𝑑} ∧ 𝑐 ≠ 𝑑)) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = inf({𝑎, 𝑏}, 𝑉, 𝑅) ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = sup({𝑎, 𝑏}, 𝑉, 𝑅)) → ((𝑐 = 𝑎 ∧ 𝑑 = 𝑏) ∨ (𝑐 = 𝑏 ∧ 𝑑 = 𝑎))))
156		vex 3454	. . . . . . . . . . . . . 14 ⊢ 𝑐 ∈ V
157		vex 3454	. . . . . . . . . . . . . 14 ⊢ 𝑑 ∈ V
158		vex 3454	. . . . . . . . . . . . . 14 ⊢ 𝑎 ∈ V
159		vex 3454	. . . . . . . . . . . . . 14 ⊢ 𝑏 ∈ V
160	156, 157, 158, 159	preq12b 4809	. . . . . . . . . . . . 13 ⊢ ({𝑐, 𝑑} = {𝑎, 𝑏} ↔ ((𝑐 = 𝑎 ∧ 𝑑 = 𝑏) ∨ (𝑐 = 𝑏 ∧ 𝑑 = 𝑎)))
161	155, 160	imbitrrdi 255	. . . . . . . . . . . 12 ⊢ (((((𝑅 Or 𝑉 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑊 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) ∧ (𝑍 = {𝑐, 𝑑} ∧ 𝑐 ≠ 𝑑)) → ((inf({𝑐, 𝑑}, 𝑉, 𝑅) = inf({𝑎, 𝑏}, 𝑉, 𝑅) ∧ sup({𝑐, 𝑑}, 𝑉, 𝑅) = sup({𝑎, 𝑏}, 𝑉, 𝑅)) → {𝑐, 𝑑} = {𝑎, 𝑏}))
162	26, 161	sylbid 243	. . . . . . . . . . 11 ⊢ (((((𝑅 Or 𝑉 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑊 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) ∧ (𝑍 = {𝑐, 𝑑} ∧ 𝑐 ≠ 𝑑)) → (⟨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 9447	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑍 = {𝑐, 𝑑} → inf(𝑍, 𝑉, 𝑅) = inf({𝑐, 𝑑}, 𝑉, 𝑅))
164		supeq1 9415	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑍 = {𝑐, 𝑑} → sup(𝑍, 𝑉, 𝑅) = sup({𝑐, 𝑑}, 𝑉, 𝑅))
165	163, 164	opeq12d 4840	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑍 = {𝑐, 𝑑} → ⟨inf(𝑍, 𝑉, 𝑅), sup(𝑍, 𝑉, 𝑅)⟩ = ⟨inf({𝑐, 𝑑}, 𝑉, 𝑅), sup({𝑐, 𝑑}, 𝑉, 𝑅)⟩)
166		infeq1 9447	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑊 = {𝑎, 𝑏} → inf(𝑊, 𝑉, 𝑅) = inf({𝑎, 𝑏}, 𝑉, 𝑅))
167		supeq1 9415	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑊 = {𝑎, 𝑏} → sup(𝑊, 𝑉, 𝑅) = sup({𝑎, 𝑏}, 𝑉, 𝑅))
168	166, 167	opeq12d 4840	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑊 = {𝑎, 𝑏} → ⟨inf(𝑊, 𝑉, 𝑅), sup(𝑊, 𝑉, 𝑅)⟩ = ⟨inf({𝑎, 𝑏}, 𝑉, 𝑅), sup({𝑎, 𝑏}, 𝑉, 𝑅)⟩)
169	165, 168	eqeqan12rd 2775	. . . . . . . . . . . . . . . . . 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 2777	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑍 = {𝑐, 𝑑} ∧ 𝑊 = {𝑎, 𝑏}) → (𝑍 = 𝑊 ↔ {𝑐, 𝑑} = {𝑎, 𝑏}))
171	170	ancoms 464	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑊 = {𝑎, 𝑏} ∧ 𝑍 = {𝑐, 𝑑}) → (𝑍 = 𝑊 ↔ {𝑐, 𝑑} = {𝑎, 𝑏}))
172	169, 171	imbi12d 347	. . . . . . . . . . . . . . . . 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})))
173	172	ex 418	. . . . . . . . . . . . . . . 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}))))
174	173	ad2antrl 741	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 Or 𝑉 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑊 = {𝑎, 𝑏} ∧ 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}))))
175	174	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 Or 𝑉 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑊 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → (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}))))
176	175	com12 33	. . . . . . . . . . . . 13 ⊢ (𝑍 = {𝑐, 𝑑} → ((((𝑅 Or 𝑉 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑊 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → ((⟨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}))))
177	176	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝑍 = {𝑐, 𝑑} ∧ 𝑐 ≠ 𝑑) → ((((𝑅 Or 𝑉 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑊 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → ((⟨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}))))
178	177	impcom 413	. . . . . . . . . . 11 ⊢ (((((𝑅 Or 𝑉 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑊 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) ∧ (𝑍 = {𝑐, 𝑑} ∧ 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})))
179	162, 178	mpbird 260	. . . . . . . . . 10 ⊢ (((((𝑅 Or 𝑉 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑊 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) ∧ (𝑍 = {𝑐, 𝑑} ∧ 𝑐 ≠ 𝑑)) → (⟨inf(𝑍, 𝑉, 𝑅), sup(𝑍, 𝑉, 𝑅)⟩ = ⟨inf(𝑊, 𝑉, 𝑅), sup(𝑊, 𝑉, 𝑅)⟩ → 𝑍 = 𝑊))
180	179	ex 418	. . . . . . . . 9 ⊢ ((((𝑅 Or 𝑉 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑊 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → ((𝑍 = {𝑐, 𝑑} ∧ 𝑐 ≠ 𝑑) → (⟨inf(𝑍, 𝑉, 𝑅), sup(𝑍, 𝑉, 𝑅)⟩ = ⟨inf(𝑊, 𝑉, 𝑅), sup(𝑊, 𝑉, 𝑅)⟩ → 𝑍 = 𝑊)))
181	180	rexlimdvva 3219	. . . . . . . 8 ⊢ (((𝑅 Or 𝑉 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑊 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)) → (∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (𝑍 = {𝑐, 𝑑} ∧ 𝑐 ≠ 𝑑) → (⟨inf(𝑍, 𝑉, 𝑅), sup(𝑍, 𝑉, 𝑅)⟩ = ⟨inf(𝑊, 𝑉, 𝑅), sup(𝑊, 𝑉, 𝑅)⟩ → 𝑍 = 𝑊)))
182	181	ex 418	. . . . . . 7 ⊢ ((𝑅 Or 𝑉 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((𝑊 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏) → (∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (𝑍 = {𝑐, 𝑑} ∧ 𝑐 ≠ 𝑑) → (⟨inf(𝑍, 𝑉, 𝑅), sup(𝑍, 𝑉, 𝑅)⟩ = ⟨inf(𝑊, 𝑉, 𝑅), sup(𝑊, 𝑉, 𝑅)⟩ → 𝑍 = 𝑊))))
183	182	rexlimdvva 3219	. . . . . 6 ⊢ (𝑅 Or 𝑉 → (∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑊 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏) → (∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (𝑍 = {𝑐, 𝑑} ∧ 𝑐 ≠ 𝑑) → (⟨inf(𝑍, 𝑉, 𝑅), sup(𝑍, 𝑉, 𝑅)⟩ = ⟨inf(𝑊, 𝑉, 𝑅), sup(𝑊, 𝑉, 𝑅)⟩ → 𝑍 = 𝑊))))
184	183	com13 89	. . . . 5 ⊢ (∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (𝑍 = {𝑐, 𝑑} ∧ 𝑐 ≠ 𝑑) → (∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑊 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏) → (𝑅 Or 𝑉 → (⟨inf(𝑍, 𝑉, 𝑅), sup(𝑍, 𝑉, 𝑅)⟩ = ⟨inf(𝑊, 𝑉, 𝑅), sup(𝑊, 𝑉, 𝑅)⟩ → 𝑍 = 𝑊))))
185	19, 184	biimtrid 245	. . . 4 ⊢ (∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (𝑍 = {𝑐, 𝑑} ∧ 𝑐 ≠ 𝑑) → (𝑊 ∈ 𝑃 → (𝑅 Or 𝑉 → (⟨inf(𝑍, 𝑉, 𝑅), sup(𝑍, 𝑉, 𝑅)⟩ = ⟨inf(𝑊, 𝑉, 𝑅), sup(𝑊, 𝑉, 𝑅)⟩ → 𝑍 = 𝑊))))
186	18, 185	sylbi 220	. . 3 ⊢ (𝑍 ∈ 𝑃 → (𝑊 ∈ 𝑃 → (𝑅 Or 𝑉 → (⟨inf(𝑍, 𝑉, 𝑅), sup(𝑍, 𝑉, 𝑅)⟩ = ⟨inf(𝑊, 𝑉, 𝑅), sup(𝑊, 𝑉, 𝑅)⟩ → 𝑍 = 𝑊))))
187	186	3imp31 1129	. 2 ⊢ ((𝑅 Or 𝑉 ∧ 𝑊 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃) → (⟨inf(𝑍, 𝑉, 𝑅), sup(𝑍, 𝑉, 𝑅)⟩ = ⟨inf(𝑊, 𝑉, 𝑅), sup(𝑊, 𝑉, 𝑅)⟩ → 𝑍 = 𝑊))
188	16, 187	sylbid 243	1 ⊢ ((𝑅 Or 𝑉 ∧ 𝑊 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃) → ((𝐹‘𝑍) = (𝐹‘𝑊) → 𝑍 = 𝑊))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∨ w3o 1102   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2955  ∃wrex 3086  {crab 3412  Vcvv 3450   ∩ cin 3897  ifcif 4481  𝒫 cpw 4556  {cpr 4585  ⟨cop 4589   class class class wbr 5102   ↦ cmpt 5185   Or wor 5554   × cxp 5645  ‘cfv 6527  supcsup 9410  infcinf 9411  2c2 12366  ♯chash 14441

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734  ax-cnex 11227  ax-resscn 11228  ax-1cn 11229  ax-icn 11230  ax-addcl 11231  ax-addrcl 11232  ax-mulcl 11233  ax-mulrcl 11234  ax-mulcom 11235  ax-addass 11236  ax-mulass 11237  ax-distr 11238  ax-i2m1 11239  ax-1ne0 11240  ax-1rid 11241  ax-rnegex 11242  ax-rrecex 11243  ax-cnre 11244  ax-pre-lttri 11245  ax-pre-lttrn 11246  ax-pre-ltadd 11247  ax-pre-mulgt0 11248

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6293  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7861  df-1st 7984  df-2nd 7985  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8454  df-2o 8455  df-oadd 8458  df-er 8695  df-en 8952  df-dom 8953  df-sdom 8954  df-fin 8955  df-sup 9412  df-inf 9413  df-dju 9953  df-card 9991  df-pnf 11316  df-mnf 11317  df-xr 11318  df-ltxr 11319  df-le 11320  df-sub 11514  df-neg 11515  df-nn 12305  df-2 12374  df-n0 12576  df-z 12663  df-uz 12935  df-fz 13609  df-hash 14442

This theorem is used by:  prproropf1o  48511