Theorem frgr3vlem2 28539

Description: Lemma 2 for frgr3v 28540. (Contributed by Alexander van der Vekens, 4-Oct-2017.) (Revised by AV, 29-Mar-2021.)

Hypotheses

Ref	Expression
frgr3v.v	⊢ 𝑉 = (Vtx‘𝐺)
frgr3v.e	⊢ 𝐸 = (Edg‘𝐺)

Assertion

Ref	Expression
frgr3vlem2	⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 ↔ ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸))))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝑥,𝐸   𝑥,𝐺   𝑥,𝑉   𝑥,𝑋   𝑥,𝑌   𝑥,𝑍

Proof of Theorem frgr3vlem2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-reu 3070	. . 3 ⊢ (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 ↔ ∃!𝑥(𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸))
2		eleq1w 2821	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑦 ∈ {𝐴, 𝐵, 𝐶}))
3		preq1 4666	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → {𝑥, 𝐴} = {𝑦, 𝐴})
4		preq1 4666	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → {𝑥, 𝐵} = {𝑦, 𝐵})
5	3, 4	preq12d 4674	. . . . . . 7 ⊢ (𝑥 = 𝑦 → {{𝑥, 𝐴}, {𝑥, 𝐵}} = {{𝑦, 𝐴}, {𝑦, 𝐵}})
6	5	sseq1d 3948	. . . . . 6 ⊢ (𝑥 = 𝑦 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 ↔ {{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸))
7	2, 6	anbi12d 630	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸) ↔ (𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸)))
8	7	eu4 2617	. . . 4 ⊢ (∃!𝑥(𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸) ↔ (∃𝑥(𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸) ∧ ∀𝑥∀𝑦(((𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸) ∧ (𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸)) → 𝑥 = 𝑦)))
9		frgr3v.v	. . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺)
10		frgr3v.e	. . . . . . . 8 ⊢ 𝐸 = (Edg‘𝐺)
11	9, 10	frgr3vlem1 28538	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ∀𝑥∀𝑦(((𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸) ∧ (𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸)) → 𝑥 = 𝑦))
12	11	3expa 1116	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ∀𝑥∀𝑦(((𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸) ∧ (𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸)) → 𝑥 = 𝑦))
13	12	biantrud 531	. . . . 5 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → (∃𝑥(𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸) ↔ (∃𝑥(𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸) ∧ ∀𝑥∀𝑦(((𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸) ∧ (𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸)) → 𝑥 = 𝑦))))
14		vex 3426	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
15	14	eltp 4621	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ∨ 𝑥 = 𝐶))
16		preq1 4666	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝐴 → {𝑥, 𝐴} = {𝐴, 𝐴})
17		preq1 4666	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝐴 → {𝑥, 𝐵} = {𝐴, 𝐵})
18	16, 17	preq12d 4674	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐴 → {{𝑥, 𝐴}, {𝑥, 𝐵}} = {{𝐴, 𝐴}, {𝐴, 𝐵}})
19	18	sseq1d 3948	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 ↔ {{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸))
20		prex 5350	. . . . . . . . . . . . . 14 ⊢ {𝐴, 𝐴} ∈ V
21		prex 5350	. . . . . . . . . . . . . 14 ⊢ {𝐴, 𝐵} ∈ V
22	20, 21	prss 4750	. . . . . . . . . . . . 13 ⊢ (({𝐴, 𝐴} ∈ 𝐸 ∧ {𝐴, 𝐵} ∈ 𝐸) ↔ {{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸)
23	10	usgredgne 27476	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺 ∈ USGraph ∧ {𝐴, 𝐴} ∈ 𝐸) → 𝐴 ≠ 𝐴)
24	23	adantll 710	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) ∧ {𝐴, 𝐴} ∈ 𝐸) → 𝐴 ≠ 𝐴)
25		df-ne 2943	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ≠ 𝐴 ↔ ¬ 𝐴 = 𝐴)
26		eqid 2738	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐴 = 𝐴
27	26	pm2.24i 150	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ 𝐴 = 𝐴 → ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸))
28	25, 27	sylbi 216	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ≠ 𝐴 → ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸))
29	24, 28	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) ∧ {𝐴, 𝐴} ∈ 𝐸) → ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸))
30	29	ex 412	. . . . . . . . . . . . . . . 16 ⊢ ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → ({𝐴, 𝐴} ∈ 𝐸 → ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)))
31	30	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ({𝐴, 𝐴} ∈ 𝐸 → ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)))
32	31	com12 32	. . . . . . . . . . . . . 14 ⊢ ({𝐴, 𝐴} ∈ 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)))
33	32	adantr 480	. . . . . . . . . . . . 13 ⊢ (({𝐴, 𝐴} ∈ 𝐸 ∧ {𝐴, 𝐵} ∈ 𝐸) → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)))
34	22, 33	sylbir 234	. . . . . . . . . . . 12 ⊢ ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)))
35	19, 34	syl6bi 252	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸))))
36		preq1 4666	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝐵 → {𝑥, 𝐴} = {𝐵, 𝐴})
37		preq1 4666	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝐵 → {𝑥, 𝐵} = {𝐵, 𝐵})
38	36, 37	preq12d 4674	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐵 → {{𝑥, 𝐴}, {𝑥, 𝐵}} = {{𝐵, 𝐴}, {𝐵, 𝐵}})
39	38	sseq1d 3948	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐵 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 ↔ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸))
40		prex 5350	. . . . . . . . . . . . . 14 ⊢ {𝐵, 𝐴} ∈ V
41		prex 5350	. . . . . . . . . . . . . 14 ⊢ {𝐵, 𝐵} ∈ V
42	40, 41	prss 4750	. . . . . . . . . . . . 13 ⊢ (({𝐵, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝐵} ∈ 𝐸) ↔ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸)
43	10	usgredgne 27476	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺 ∈ USGraph ∧ {𝐵, 𝐵} ∈ 𝐸) → 𝐵 ≠ 𝐵)
44	43	adantll 710	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) ∧ {𝐵, 𝐵} ∈ 𝐸) → 𝐵 ≠ 𝐵)
45		df-ne 2943	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵 ≠ 𝐵 ↔ ¬ 𝐵 = 𝐵)
46		eqid 2738	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐵 = 𝐵
47	46	pm2.24i 150	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ 𝐵 = 𝐵 → ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸))
48	45, 47	sylbi 216	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ≠ 𝐵 → ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸))
49	44, 48	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) ∧ {𝐵, 𝐵} ∈ 𝐸) → ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸))
50	49	ex 412	. . . . . . . . . . . . . . . 16 ⊢ ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → ({𝐵, 𝐵} ∈ 𝐸 → ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)))
51	50	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ({𝐵, 𝐵} ∈ 𝐸 → ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)))
52	51	com12 32	. . . . . . . . . . . . . 14 ⊢ ({𝐵, 𝐵} ∈ 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)))
53	52	adantl 481	. . . . . . . . . . . . 13 ⊢ (({𝐵, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝐵} ∈ 𝐸) → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)))
54	42, 53	sylbir 234	. . . . . . . . . . . 12 ⊢ ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)))
55	39, 54	syl6bi 252	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐵 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸))))
56		preq1 4666	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝐶 → {𝑥, 𝐴} = {𝐶, 𝐴})
57		preq1 4666	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝐶 → {𝑥, 𝐵} = {𝐶, 𝐵})
58	56, 57	preq12d 4674	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐶 → {{𝑥, 𝐴}, {𝑥, 𝐵}} = {{𝐶, 𝐴}, {𝐶, 𝐵}})
59	58	sseq1d 3948	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐶 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 ↔ {{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ 𝐸))
60		prex 5350	. . . . . . . . . . . . . 14 ⊢ {𝐶, 𝐴} ∈ V
61		prex 5350	. . . . . . . . . . . . . 14 ⊢ {𝐶, 𝐵} ∈ V
62	60, 61	prss 4750	. . . . . . . . . . . . 13 ⊢ (({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸) ↔ {{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ 𝐸)
63		ax-1 6	. . . . . . . . . . . . 13 ⊢ (({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸) → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)))
64	62, 63	sylbir 234	. . . . . . . . . . . 12 ⊢ ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)))
65	59, 64	syl6bi 252	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐶 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸))))
66	35, 55, 65	3jaoi 1425	. . . . . . . . . 10 ⊢ ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ∨ 𝑥 = 𝐶) → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸))))
67	15, 66	sylbi 216	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝐴, 𝐵, 𝐶} → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸))))
68	67	imp 406	. . . . . . . 8 ⊢ ((𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸) → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)))
69	68	com12 32	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ((𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸) → ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)))
70	69	exlimdv 1937	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → (∃𝑥(𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸) → ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)))
71		prssi 4751	. . . . . . . . . . 11 ⊢ (({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸) → {{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ 𝐸)
72	71	adantl 481	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)) → {{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ 𝐸)
73	72	3mix3d 1336	. . . . . . . . 9 ⊢ (((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)) → ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 ∨ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 ∨ {{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ 𝐸))
74	19, 39, 59	rextpg 4632	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → (∃𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 ↔ ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 ∨ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 ∨ {{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ 𝐸)))
75	74	ad3antrrr 726	. . . . . . . . 9 ⊢ (((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)) → (∃𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 ↔ ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 ∨ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 ∨ {{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ 𝐸)))
76	73, 75	mpbird 256	. . . . . . . 8 ⊢ (((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)) → ∃𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸)
77		df-rex 3069	. . . . . . . 8 ⊢ (∃𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 ↔ ∃𝑥(𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸))
78	76, 77	sylib 217	. . . . . . 7 ⊢ (((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)) → ∃𝑥(𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸))
79	78	ex 412	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → (({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸) → ∃𝑥(𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸)))
80	70, 79	impbid 211	. . . . 5 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → (∃𝑥(𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸) ↔ ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)))
81	13, 80	bitr3d 280	. . . 4 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ((∃𝑥(𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸) ∧ ∀𝑥∀𝑦(((𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸) ∧ (𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸)) → 𝑥 = 𝑦)) ↔ ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)))
82	8, 81	syl5bb 282	. . 3 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → (∃!𝑥(𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸) ↔ ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)))
83	1, 82	syl5bb 282	. 2 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 ↔ ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)))
84	83	ex 412	1 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 ↔ ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∨ w3o 1084   ∧ w3a 1085  ∀wal 1537   = wceq 1539  ∃wex 1783   ∈ wcel 2108  ∃!weu 2568   ≠ wne 2942  ∃wrex 3064  ∃!wreu 3065   ⊆ wss 3883  {cpr 4560  {ctp 4562  ‘cfv 6418  Vtxcvtx 27269  Edgcedg 27320  USGraphcusgr 27422

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-oadd 8271  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-dju 9590  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-n0 12164  df-z 12250  df-uz 12512  df-fz 13169  df-hash 13973  df-edg 27321  df-umgr 27356  df-usgr 27424

This theorem is referenced by:  frgr3v  28540