Theorem frgr3vlem1 30202

Description: Lemma 1 for frgr3v 30204. (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
frgr3vlem1	⊢ (((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))

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

Proof of Theorem frgr3vlem1

Step	Hyp	Ref	Expression
1		vex 3451	. . . . . 6 ⊢ 𝑥 ∈ V
2	1	eltp 4653	. . . . 5 ⊢ (𝑥 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ∨ 𝑥 = 𝐶))
3		vex 3451	. . . . . . . . 9 ⊢ 𝑦 ∈ V
4	3	eltp 4653	. . . . . . . 8 ⊢ (𝑦 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝑦 = 𝐴 ∨ 𝑦 = 𝐵 ∨ 𝑦 = 𝐶))
5		eqidd 2730	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐴 = 𝐴)
6	5	a1i 11	. . . . . . . . . . . . . 14 ⊢ ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐴 = 𝐴))
7	6	a1i13 27	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐴 → ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 → ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐴 = 𝐴))))
8		preq1 4697	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝐴 → {𝑦, 𝐴} = {𝐴, 𝐴})
9		preq1 4697	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝐴 → {𝑦, 𝐵} = {𝐴, 𝐵})
10	8, 9	preq12d 4705	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝐴 → {{𝑦, 𝐴}, {𝑦, 𝐵}} = {{𝐴, 𝐴}, {𝐴, 𝐵}})
11	10	sseq1d 3978	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐴 → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸 ↔ {{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸))
12		eqeq2 2741	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝐴 → (𝐴 = 𝑦 ↔ 𝐴 = 𝐴))
13	12	imbi2d 340	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝐴 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐴 = 𝑦) ↔ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐴 = 𝐴)))
14	13	imbi2d 340	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐴 → (({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐴 = 𝑦)) ↔ ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐴 = 𝐴))))
15	7, 11, 14	3imtr4d 294	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐴 → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸 → ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐴 = 𝑦))))
16		prex 5392	. . . . . . . . . . . . . . . . . . 19 ⊢ {𝐴, 𝐴} ∈ V
17		prex 5392	. . . . . . . . . . . . . . . . . . 19 ⊢ {𝐴, 𝐵} ∈ V
18	16, 17	prss 4784	. . . . . . . . . . . . . . . . . 18 ⊢ (({𝐴, 𝐴} ∈ 𝐸 ∧ {𝐴, 𝐵} ∈ 𝐸) ↔ {{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸)
19		frgr3v.e	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝐸 = (Edg‘𝐺)
20	19	usgredgne 29133	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐺 ∈ USGraph ∧ {𝐴, 𝐴} ∈ 𝐸) → 𝐴 ≠ 𝐴)
21	20	adantll 714	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) ∧ {𝐴, 𝐴} ∈ 𝐸) → 𝐴 ≠ 𝐴)
22		df-ne 2926	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐴 ≠ 𝐴 ↔ ¬ 𝐴 = 𝐴)
23		eqid 2729	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝐴 = 𝐴
24	23	pm2.24i 150	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (¬ 𝐴 = 𝐴 → 𝐴 = 𝐵)
25	22, 24	sylbi 217	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴 ≠ 𝐴 → 𝐴 = 𝐵)
26	21, 25	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) ∧ {𝐴, 𝐴} ∈ 𝐸) → 𝐴 = 𝐵)
27	26	expcom 413	. . . . . . . . . . . . . . . . . . 19 ⊢ ({𝐴, 𝐴} ∈ 𝐸 → ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → 𝐴 = 𝐵))
28	27	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (({𝐴, 𝐴} ∈ 𝐸 ∧ {𝐴, 𝐵} ∈ 𝐸) → ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → 𝐴 = 𝐵))
29	18, 28	sylbir 235	. . . . . . . . . . . . . . . . 17 ⊢ ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 → ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → 𝐴 = 𝐵))
30	29	com12 32	. . . . . . . . . . . . . . . 16 ⊢ ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 → 𝐴 = 𝐵))
31	30	3ad2ant3 1135	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 → 𝐴 = 𝐵))
32	31	com12 32	. . . . . . . . . . . . . 14 ⊢ ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐴 = 𝐵))
33	32	2a1i 12	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐵 → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 → ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐴 = 𝐵))))
34		preq1 4697	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝐵 → {𝑦, 𝐴} = {𝐵, 𝐴})
35		preq1 4697	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝐵 → {𝑦, 𝐵} = {𝐵, 𝐵})
36	34, 35	preq12d 4705	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝐵 → {{𝑦, 𝐴}, {𝑦, 𝐵}} = {{𝐵, 𝐴}, {𝐵, 𝐵}})
37	36	sseq1d 3978	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐵 → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸 ↔ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸))
38		eqeq2 2741	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝐵 → (𝐴 = 𝑦 ↔ 𝐴 = 𝐵))
39	38	imbi2d 340	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝐵 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐴 = 𝑦) ↔ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐴 = 𝐵)))
40	39	imbi2d 340	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐵 → (({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐴 = 𝑦)) ↔ ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐴 = 𝐵))))
41	33, 37, 40	3imtr4d 294	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐵 → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸 → ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐴 = 𝑦))))
42	23	pm2.24i 150	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (¬ 𝐴 = 𝐴 → 𝐴 = 𝐶)
43	22, 42	sylbi 217	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴 ≠ 𝐴 → 𝐴 = 𝐶)
44	21, 43	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) ∧ {𝐴, 𝐴} ∈ 𝐸) → 𝐴 = 𝐶)
45	44	expcom 413	. . . . . . . . . . . . . . . . . . 19 ⊢ ({𝐴, 𝐴} ∈ 𝐸 → ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → 𝐴 = 𝐶))
46	45	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (({𝐴, 𝐴} ∈ 𝐸 ∧ {𝐴, 𝐵} ∈ 𝐸) → ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → 𝐴 = 𝐶))
47	18, 46	sylbir 235	. . . . . . . . . . . . . . . . 17 ⊢ ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 → ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → 𝐴 = 𝐶))
48	47	com12 32	. . . . . . . . . . . . . . . 16 ⊢ ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 → 𝐴 = 𝐶))
49	48	3ad2ant3 1135	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 → 𝐴 = 𝐶))
50	49	com12 32	. . . . . . . . . . . . . 14 ⊢ ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐴 = 𝐶))
51	50	2a1i 12	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐶 → ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ 𝐸 → ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐴 = 𝐶))))
52		preq1 4697	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝐶 → {𝑦, 𝐴} = {𝐶, 𝐴})
53		preq1 4697	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝐶 → {𝑦, 𝐵} = {𝐶, 𝐵})
54	52, 53	preq12d 4705	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝐶 → {{𝑦, 𝐴}, {𝑦, 𝐵}} = {{𝐶, 𝐴}, {𝐶, 𝐵}})
55	54	sseq1d 3978	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐶 → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸 ↔ {{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ 𝐸))
56		eqeq2 2741	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝐶 → (𝐴 = 𝑦 ↔ 𝐴 = 𝐶))
57	56	imbi2d 340	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝐶 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐴 = 𝑦) ↔ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐴 = 𝐶)))
58	57	imbi2d 340	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐶 → (({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐴 = 𝑦)) ↔ ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐴 = 𝐶))))
59	51, 55, 58	3imtr4d 294	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐶 → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸 → ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐴 = 𝑦))))
60	15, 41, 59	3jaoi 1430	. . . . . . . . . . 11 ⊢ ((𝑦 = 𝐴 ∨ 𝑦 = 𝐵 ∨ 𝑦 = 𝐶) → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸 → ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐴 = 𝑦))))
61		preq1 4697	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝐴 → {𝑥, 𝐴} = {𝐴, 𝐴})
62		preq1 4697	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝐴 → {𝑥, 𝐵} = {𝐴, 𝐵})
63	61, 62	preq12d 4705	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝐴 → {{𝑥, 𝐴}, {𝑥, 𝐵}} = {{𝐴, 𝐴}, {𝐴, 𝐵}})
64	63	sseq1d 3978	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐴 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 ↔ {{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸))
65		eqeq1 2733	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝑦 ↔ 𝐴 = 𝑦))
66	65	imbi2d 340	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐴 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝑥 = 𝑦) ↔ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐴 = 𝑦)))
67	64, 66	imbi12d 344	. . . . . . . . . . . 12 ⊢ (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)) → 𝑥 = 𝑦)) ↔ ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐴 = 𝑦))))
68	67	imbi2d 340	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → (({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝑥 = 𝑦))) ↔ ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸 → ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐴 = 𝑦)))))
69	60, 68	imbitrrid 246	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → ((𝑦 = 𝐴 ∨ 𝑦 = 𝐵 ∨ 𝑦 = 𝐶) → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝑥 = 𝑦)))))
70		prex 5392	. . . . . . . . . . . . . . . . . . 19 ⊢ {𝐵, 𝐴} ∈ V
71		prex 5392	. . . . . . . . . . . . . . . . . . 19 ⊢ {𝐵, 𝐵} ∈ V
72	70, 71	prss 4784	. . . . . . . . . . . . . . . . . 18 ⊢ (({𝐵, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝐵} ∈ 𝐸) ↔ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸)
73	19	usgredgne 29133	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐺 ∈ USGraph ∧ {𝐵, 𝐵} ∈ 𝐸) → 𝐵 ≠ 𝐵)
74	73	adantll 714	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) ∧ {𝐵, 𝐵} ∈ 𝐸) → 𝐵 ≠ 𝐵)
75		df-ne 2926	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐵 ≠ 𝐵 ↔ ¬ 𝐵 = 𝐵)
76		eqid 2729	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝐵 = 𝐵
77	76	pm2.24i 150	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (¬ 𝐵 = 𝐵 → 𝐵 = 𝐴)
78	75, 77	sylbi 217	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐵 ≠ 𝐵 → 𝐵 = 𝐴)
79	74, 78	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) ∧ {𝐵, 𝐵} ∈ 𝐸) → 𝐵 = 𝐴)
80	79	expcom 413	. . . . . . . . . . . . . . . . . . 19 ⊢ ({𝐵, 𝐵} ∈ 𝐸 → ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → 𝐵 = 𝐴))
81	80	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ (({𝐵, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝐵} ∈ 𝐸) → ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → 𝐵 = 𝐴))
82	72, 81	sylbir 235	. . . . . . . . . . . . . . . . 17 ⊢ ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 → ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → 𝐵 = 𝐴))
83	82	com12 32	. . . . . . . . . . . . . . . 16 ⊢ ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 → 𝐵 = 𝐴))
84	83	3ad2ant3 1135	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 → 𝐵 = 𝐴))
85	84	com12 32	. . . . . . . . . . . . . 14 ⊢ ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐵 = 𝐴))
86	85	2a1i 12	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐴 → ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐵 = 𝐴))))
87		eqeq2 2741	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝐴 → (𝐵 = 𝑦 ↔ 𝐵 = 𝐴))
88	87	imbi2d 340	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝐴 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐵 = 𝑦) ↔ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐵 = 𝐴)))
89	88	imbi2d 340	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐴 → (({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐵 = 𝑦)) ↔ ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐵 = 𝐴))))
90	86, 11, 89	3imtr4d 294	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐴 → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸 → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐵 = 𝑦))))
91		eqidd 2730	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐵 = 𝐵)
92	91	a1i 11	. . . . . . . . . . . . . 14 ⊢ ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐵 = 𝐵))
93	92	a1i13 27	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐵 → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐵 = 𝐵))))
94		eqeq2 2741	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝐵 → (𝐵 = 𝑦 ↔ 𝐵 = 𝐵))
95	94	imbi2d 340	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝐵 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐵 = 𝑦) ↔ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐵 = 𝐵)))
96	95	imbi2d 340	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐵 → (({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐵 = 𝑦)) ↔ ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐵 = 𝐵))))
97	93, 37, 96	3imtr4d 294	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐵 → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸 → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐵 = 𝑦))))
98	76	pm2.24i 150	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (¬ 𝐵 = 𝐵 → 𝐵 = 𝐶)
99	75, 98	sylbi 217	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐵 ≠ 𝐵 → 𝐵 = 𝐶)
100	74, 99	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) ∧ {𝐵, 𝐵} ∈ 𝐸) → 𝐵 = 𝐶)
101	100	expcom 413	. . . . . . . . . . . . . . . . . . 19 ⊢ ({𝐵, 𝐵} ∈ 𝐸 → ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → 𝐵 = 𝐶))
102	101	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ (({𝐵, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝐵} ∈ 𝐸) → ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → 𝐵 = 𝐶))
103	72, 102	sylbir 235	. . . . . . . . . . . . . . . . 17 ⊢ ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 → ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → 𝐵 = 𝐶))
104	103	com12 32	. . . . . . . . . . . . . . . 16 ⊢ ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 → 𝐵 = 𝐶))
105	104	3ad2ant3 1135	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 → 𝐵 = 𝐶))
106	105	com12 32	. . . . . . . . . . . . . 14 ⊢ ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐵 = 𝐶))
107	106	2a1i 12	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐶 → ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ 𝐸 → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐵 = 𝐶))))
108		eqeq2 2741	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝐶 → (𝐵 = 𝑦 ↔ 𝐵 = 𝐶))
109	108	imbi2d 340	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝐶 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐵 = 𝑦) ↔ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐵 = 𝐶)))
110	109	imbi2d 340	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐶 → (({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐵 = 𝑦)) ↔ ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐵 = 𝐶))))
111	107, 55, 110	3imtr4d 294	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐶 → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸 → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐵 = 𝑦))))
112	90, 97, 111	3jaoi 1430	. . . . . . . . . . 11 ⊢ ((𝑦 = 𝐴 ∨ 𝑦 = 𝐵 ∨ 𝑦 = 𝐶) → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸 → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐵 = 𝑦))))
113		preq1 4697	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝐵 → {𝑥, 𝐴} = {𝐵, 𝐴})
114		preq1 4697	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝐵 → {𝑥, 𝐵} = {𝐵, 𝐵})
115	113, 114	preq12d 4705	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝐵 → {{𝑥, 𝐴}, {𝑥, 𝐵}} = {{𝐵, 𝐴}, {𝐵, 𝐵}})
116	115	sseq1d 3978	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐵 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 ↔ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸))
117		eqeq1 2733	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝐵 → (𝑥 = 𝑦 ↔ 𝐵 = 𝑦))
118	117	imbi2d 340	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐵 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝑥 = 𝑦) ↔ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐵 = 𝑦)))
119	116, 118	imbi12d 344	. . . . . . . . . . . 12 ⊢ (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)) → 𝑥 = 𝑦)) ↔ ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐵 = 𝑦))))
120	119	imbi2d 340	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐵 → (({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝑥 = 𝑦))) ↔ ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸 → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐵 = 𝑦)))))
121	112, 120	imbitrrid 246	. . . . . . . . . 10 ⊢ (𝑥 = 𝐵 → ((𝑦 = 𝐴 ∨ 𝑦 = 𝐵 ∨ 𝑦 = 𝐶) → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝑥 = 𝑦)))))
122	23	pm2.24i 150	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (¬ 𝐴 = 𝐴 → 𝐶 = 𝐴)
123	22, 122	sylbi 217	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴 ≠ 𝐴 → 𝐶 = 𝐴)
124	21, 123	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) ∧ {𝐴, 𝐴} ∈ 𝐸) → 𝐶 = 𝐴)
125	124	expcom 413	. . . . . . . . . . . . . . . . . . 19 ⊢ ({𝐴, 𝐴} ∈ 𝐸 → ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → 𝐶 = 𝐴))
126	125	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (({𝐴, 𝐴} ∈ 𝐸 ∧ {𝐴, 𝐵} ∈ 𝐸) → ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → 𝐶 = 𝐴))
127	18, 126	sylbir 235	. . . . . . . . . . . . . . . . 17 ⊢ ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 → ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → 𝐶 = 𝐴))
128	127	com12 32	. . . . . . . . . . . . . . . 16 ⊢ ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 → 𝐶 = 𝐴))
129	128	3ad2ant3 1135	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 → 𝐶 = 𝐴))
130	129	com12 32	. . . . . . . . . . . . . 14 ⊢ ({{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))
131	130	a1i13 27	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐴 → ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 → ({{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))))
132		eqeq2 2741	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝐴 → (𝐶 = 𝑦 ↔ 𝐶 = 𝐴))
133	132	imbi2d 340	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝐴 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐶 = 𝑦) ↔ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐶 = 𝐴)))
134	133	imbi2d 340	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐴 → (({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ 𝐸 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐶 = 𝑦)) ↔ ({{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))))
135	131, 11, 134	3imtr4d 294	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐴 → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸 → ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ 𝐸 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐶 = 𝑦))))
136		pm2.21 123	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (¬ 𝐵 = 𝐵 → (𝐵 = 𝐵 → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → 𝐶 = 𝐵)))
137	75, 136	sylbi 217	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐵 ≠ 𝐵 → (𝐵 = 𝐵 → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → 𝐶 = 𝐵)))
138	74, 76, 137	mpisyl 21	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) ∧ {𝐵, 𝐵} ∈ 𝐸) → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → 𝐶 = 𝐵))
139	138	expcom 413	. . . . . . . . . . . . . . . . . . . 20 ⊢ ({𝐵, 𝐵} ∈ 𝐸 → ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → 𝐶 = 𝐵)))
140	139	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (({𝐵, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝐵} ∈ 𝐸) → ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → 𝐶 = 𝐵)))
141	72, 140	sylbir 235	. . . . . . . . . . . . . . . . . 18 ⊢ ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 → ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → 𝐶 = 𝐵)))
142	141	com13 88	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 → 𝐶 = 𝐵)))
143	142	a1d 25	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 → 𝐶 = 𝐵))))
144	143	3imp 1110	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 → 𝐶 = 𝐵))
145	144	com12 32	. . . . . . . . . . . . . 14 ⊢ ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐶 = 𝐵))
146	145	a1i13 27	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐵 → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 → ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ 𝐸 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐶 = 𝐵))))
147		eqeq2 2741	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝐵 → (𝐶 = 𝑦 ↔ 𝐶 = 𝐵))
148	147	imbi2d 340	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝐵 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐶 = 𝑦) ↔ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐶 = 𝐵)))
149	148	imbi2d 340	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐵 → (({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ 𝐸 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐶 = 𝑦)) ↔ ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ 𝐸 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐶 = 𝐵))))
150	146, 37, 149	3imtr4d 294	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐵 → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸 → ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ 𝐸 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐶 = 𝑦))))
151		eqidd 2730	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐶 = 𝐶)
152	151	a1i 11	. . . . . . . . . . . . . 14 ⊢ ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ 𝐸 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐶 = 𝐶))
153	152	a1i13 27	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐶 → ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ 𝐸 → ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ 𝐸 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐶 = 𝐶))))
154		eqeq2 2741	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝐶 → (𝐶 = 𝑦 ↔ 𝐶 = 𝐶))
155	154	imbi2d 340	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝐶 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐶 = 𝑦) ↔ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐶 = 𝐶)))
156	155	imbi2d 340	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐶 → (({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ 𝐸 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐶 = 𝑦)) ↔ ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ 𝐸 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐶 = 𝐶))))
157	153, 55, 156	3imtr4d 294	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐶 → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸 → ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ 𝐸 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐶 = 𝑦))))
158	135, 150, 157	3jaoi 1430	. . . . . . . . . . 11 ⊢ ((𝑦 = 𝐴 ∨ 𝑦 = 𝐵 ∨ 𝑦 = 𝐶) → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸 → ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ 𝐸 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐶 = 𝑦))))
159		preq1 4697	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝐶 → {𝑥, 𝐴} = {𝐶, 𝐴})
160		preq1 4697	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝐶 → {𝑥, 𝐵} = {𝐶, 𝐵})
161	159, 160	preq12d 4705	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝐶 → {{𝑥, 𝐴}, {𝑥, 𝐵}} = {{𝐶, 𝐴}, {𝐶, 𝐵}})
162	161	sseq1d 3978	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐶 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 ↔ {{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ 𝐸))
163		eqeq1 2733	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝐶 → (𝑥 = 𝑦 ↔ 𝐶 = 𝑦))
164	163	imbi2d 340	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐶 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝑥 = 𝑦) ↔ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐶 = 𝑦)))
165	162, 164	imbi12d 344	. . . . . . . . . . . 12 ⊢ (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)) → 𝑥 = 𝑦)) ↔ ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ 𝐸 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐶 = 𝑦))))
166	165	imbi2d 340	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐶 → (({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝑥 = 𝑦))) ↔ ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸 → ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ 𝐸 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐶 = 𝑦)))))
167	158, 166	imbitrrid 246	. . . . . . . . . 10 ⊢ (𝑥 = 𝐶 → ((𝑦 = 𝐴 ∨ 𝑦 = 𝐵 ∨ 𝑦 = 𝐶) → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝑥 = 𝑦)))))
168	69, 121, 167	3jaoi 1430	. . . . . . . . 9 ⊢ ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ∨ 𝑥 = 𝐶) → ((𝑦 = 𝐴 ∨ 𝑦 = 𝐵 ∨ 𝑦 = 𝐶) → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝑥 = 𝑦)))))
169	168	com3l 89	. . . . . . . 8 ⊢ ((𝑦 = 𝐴 ∨ 𝑦 = 𝐵 ∨ 𝑦 = 𝐶) → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸 → ((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)) → 𝑥 = 𝑦)))))
170	4, 169	sylbi 217	. . . . . . 7 ⊢ (𝑦 ∈ {𝐴, 𝐵, 𝐶} → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸 → ((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)) → 𝑥 = 𝑦)))))
171	170	imp 406	. . . . . 6 ⊢ ((𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸) → ((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)) → 𝑥 = 𝑦))))
172	171	com3l 89	. . . . 5 ⊢ ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ∨ 𝑥 = 𝐶) → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 → ((𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸) → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝑥 = 𝑦))))
173	2, 172	sylbi 217	. . . 4 ⊢ (𝑥 ∈ {𝐴, 𝐵, 𝐶} → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 → ((𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸) → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝑥 = 𝑦))))
174	173	imp31 417	. . 3 ⊢ (((𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸) ∧ (𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸)) → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝑥 = 𝑦))
175	174	com12 32	. 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) ∧ (𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸)) → 𝑥 = 𝑦))
176	175	alrimivv 1928	1 ⊢ (((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))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ w3o 1085   ∧ w3a 1086  ∀wal 1538   = wceq 1540   ∈ wcel 2109   ≠ wne 2925   ⊆ wss 3914  {cpr 4591  {ctp 4593  ‘cfv 6511  Vtxcvtx 28923  Edgcedg 28974  USGraphcusgr 29076

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-oadd 8438  df-er 8671  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-dju 9854  df-card 9892  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-2 12249  df-n0 12443  df-z 12530  df-uz 12794  df-fz 13469  df-hash 14296  df-edg 28975  df-umgr 29010  df-usgr 29078

This theorem is referenced by:  frgr3vlem2  30203