Theorem frgra3vlem1 26527

Description: Lemma 1 for frgra3v 26529. (Contributed by Alexander van der Vekens, 4-Oct-2017.)

Assertion

Ref	Expression
frgra3vlem1	⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → ∀𝑥∀𝑦(((𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸) ∧ (𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸)) → 𝑥 = 𝑦))

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

Proof of Theorem frgra3vlem1

Step	Hyp	Ref	Expression
1		vex 3176	. . . . . 6 ⊢ 𝑥 ∈ V
2	1	eltp 4177	. . . . 5 ⊢ (𝑥 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ∨ 𝑥 = 𝐶))
3		vex 3176	. . . . . . . . 9 ⊢ 𝑦 ∈ V
4	3	eltp 4177	. . . . . . . 8 ⊢ (𝑦 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝑦 = 𝐴 ∨ 𝑦 = 𝐵 ∨ 𝑦 = 𝐶))
5		eqidd 2611	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐴 = 𝐴)
6	5	a1i 11	. . . . . . . . . . . . . 14 ⊢ ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐴 = 𝐴))
7	6	2a1i 12	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐴 → ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 → ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐴 = 𝐴))))
8		preq1 4212	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝐴 → {𝑦, 𝐴} = {𝐴, 𝐴})
9		preq1 4212	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝐴 → {𝑦, 𝐵} = {𝐴, 𝐵})
10	8, 9	preq12d 4220	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝐴 → {{𝑦, 𝐴}, {𝑦, 𝐵}} = {{𝐴, 𝐴}, {𝐴, 𝐵}})
11	10	sseq1d 3595	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐴 → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸 ↔ {{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸))
12		eqeq2 2621	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝐴 → (𝐴 = 𝑦 ↔ 𝐴 = 𝐴))
13	12	imbi2d 329	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝐴 → (((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐴 = 𝑦) ↔ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐴 = 𝐴)))
14	13	imbi2d 329	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐴 → (({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐴 = 𝑦)) ↔ ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐴 = 𝐴))))
15	7, 11, 14	3imtr4d 282	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐴 → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸 → ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐴 = 𝑦))))
16		prex 4836	. . . . . . . . . . . . . . . . 17 ⊢ {𝐴, 𝐴} ∈ V
17		prex 4836	. . . . . . . . . . . . . . . . 17 ⊢ {𝐴, 𝐵} ∈ V
18	16, 17	prss 4291	. . . . . . . . . . . . . . . 16 ⊢ (({𝐴, 𝐴} ∈ ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸) ↔ {{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸)
19		usgraedgrn 25910	. . . . . . . . . . . . . . . . . . 19 ⊢ (({𝐴, 𝐵, 𝐶} USGrph 𝐸 ∧ {𝐴, 𝐴} ∈ ran 𝐸) → 𝐴 ≠ 𝐴)
20		df-ne 2782	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ≠ 𝐴 ↔ ¬ 𝐴 = 𝐴)
21		eqid 2610	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝐴 = 𝐴
22	21	pm2.24i 145	. . . . . . . . . . . . . . . . . . . 20 ⊢ (¬ 𝐴 = 𝐴 → 𝐴 = 𝐵)
23	20, 22	sylbi 206	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ≠ 𝐴 → 𝐴 = 𝐵)
24	19, 23	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (({𝐴, 𝐵, 𝐶} USGrph 𝐸 ∧ {𝐴, 𝐴} ∈ ran 𝐸) → 𝐴 = 𝐵)
25	24	expcom 450	. . . . . . . . . . . . . . . . 17 ⊢ ({𝐴, 𝐴} ∈ ran 𝐸 → ({𝐴, 𝐵, 𝐶} USGrph 𝐸 → 𝐴 = 𝐵))
26	25	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (({𝐴, 𝐴} ∈ ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸) → ({𝐴, 𝐵, 𝐶} USGrph 𝐸 → 𝐴 = 𝐵))
27	18, 26	sylbir 224	. . . . . . . . . . . . . . 15 ⊢ ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 → ({𝐴, 𝐵, 𝐶} USGrph 𝐸 → 𝐴 = 𝐵))
28	27	adantld 482	. . . . . . . . . . . . . 14 ⊢ ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐴 = 𝐵))
29	28	2a1i 12	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐵 → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸 → ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐴 = 𝐵))))
30		preq1 4212	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝐵 → {𝑦, 𝐴} = {𝐵, 𝐴})
31		preq1 4212	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝐵 → {𝑦, 𝐵} = {𝐵, 𝐵})
32	30, 31	preq12d 4220	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝐵 → {{𝑦, 𝐴}, {𝑦, 𝐵}} = {{𝐵, 𝐴}, {𝐵, 𝐵}})
33	32	sseq1d 3595	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐵 → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸 ↔ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸))
34		eqeq2 2621	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝐵 → (𝐴 = 𝑦 ↔ 𝐴 = 𝐵))
35	34	imbi2d 329	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝐵 → (((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐴 = 𝑦) ↔ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐴 = 𝐵)))
36	35	imbi2d 329	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐵 → (({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐴 = 𝑦)) ↔ ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐴 = 𝐵))))
37	29, 33, 36	3imtr4d 282	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐵 → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸 → ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐴 = 𝑦))))
38	21	pm2.24i 145	. . . . . . . . . . . . . . . . . . . 20 ⊢ (¬ 𝐴 = 𝐴 → 𝐴 = 𝐶)
39	20, 38	sylbi 206	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ≠ 𝐴 → 𝐴 = 𝐶)
40	19, 39	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (({𝐴, 𝐵, 𝐶} USGrph 𝐸 ∧ {𝐴, 𝐴} ∈ ran 𝐸) → 𝐴 = 𝐶)
41	40	expcom 450	. . . . . . . . . . . . . . . . 17 ⊢ ({𝐴, 𝐴} ∈ ran 𝐸 → ({𝐴, 𝐵, 𝐶} USGrph 𝐸 → 𝐴 = 𝐶))
42	41	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (({𝐴, 𝐴} ∈ ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸) → ({𝐴, 𝐵, 𝐶} USGrph 𝐸 → 𝐴 = 𝐶))
43	18, 42	sylbir 224	. . . . . . . . . . . . . . 15 ⊢ ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 → ({𝐴, 𝐵, 𝐶} USGrph 𝐸 → 𝐴 = 𝐶))
44	43	adantld 482	. . . . . . . . . . . . . 14 ⊢ ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐴 = 𝐶))
45	44	2a1i 12	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐶 → ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ ran 𝐸 → ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐴 = 𝐶))))
46		preq1 4212	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝐶 → {𝑦, 𝐴} = {𝐶, 𝐴})
47		preq1 4212	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝐶 → {𝑦, 𝐵} = {𝐶, 𝐵})
48	46, 47	preq12d 4220	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝐶 → {{𝑦, 𝐴}, {𝑦, 𝐵}} = {{𝐶, 𝐴}, {𝐶, 𝐵}})
49	48	sseq1d 3595	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐶 → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸 ↔ {{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ ran 𝐸))
50		eqeq2 2621	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝐶 → (𝐴 = 𝑦 ↔ 𝐴 = 𝐶))
51	50	imbi2d 329	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝐶 → (((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐴 = 𝑦) ↔ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐴 = 𝐶)))
52	51	imbi2d 329	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐶 → (({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐴 = 𝑦)) ↔ ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐴 = 𝐶))))
53	45, 49, 52	3imtr4d 282	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐶 → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸 → ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐴 = 𝑦))))
54	15, 37, 53	3jaoi 1383	. . . . . . . . . . 11 ⊢ ((𝑦 = 𝐴 ∨ 𝑦 = 𝐵 ∨ 𝑦 = 𝐶) → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸 → ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐴 = 𝑦))))
55		preq1 4212	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝐴 → {𝑥, 𝐴} = {𝐴, 𝐴})
56		preq1 4212	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝐴 → {𝑥, 𝐵} = {𝐴, 𝐵})
57	55, 56	preq12d 4220	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝐴 → {{𝑥, 𝐴}, {𝑥, 𝐵}} = {{𝐴, 𝐴}, {𝐴, 𝐵}})
58	57	sseq1d 3595	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐴 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 ↔ {{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸))
59		eqeq1 2614	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝑦 ↔ 𝐴 = 𝑦))
60	59	imbi2d 329	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐴 → (((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝑥 = 𝑦) ↔ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐴 = 𝑦)))
61	58, 60	imbi12d 333	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → (({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝑥 = 𝑦)) ↔ ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐴 = 𝑦))))
62	61	imbi2d 329	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → (({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝑥 = 𝑦))) ↔ ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸 → ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐴 = 𝑦)))))
63	54, 62	syl5ibr 235	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → ((𝑦 = 𝐴 ∨ 𝑦 = 𝐵 ∨ 𝑦 = 𝐶) → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝑥 = 𝑦)))))
64		prex 4836	. . . . . . . . . . . . . . . . 17 ⊢ {𝐵, 𝐴} ∈ V
65		prex 4836	. . . . . . . . . . . . . . . . 17 ⊢ {𝐵, 𝐵} ∈ V
66	64, 65	prss 4291	. . . . . . . . . . . . . . . 16 ⊢ (({𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐵, 𝐵} ∈ ran 𝐸) ↔ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸)
67		usgraedgrn 25910	. . . . . . . . . . . . . . . . . . 19 ⊢ (({𝐴, 𝐵, 𝐶} USGrph 𝐸 ∧ {𝐵, 𝐵} ∈ ran 𝐸) → 𝐵 ≠ 𝐵)
68		df-ne 2782	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐵 ≠ 𝐵 ↔ ¬ 𝐵 = 𝐵)
69		eqid 2610	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝐵 = 𝐵
70	69	pm2.24i 145	. . . . . . . . . . . . . . . . . . . 20 ⊢ (¬ 𝐵 = 𝐵 → 𝐵 = 𝐴)
71	68, 70	sylbi 206	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵 ≠ 𝐵 → 𝐵 = 𝐴)
72	67, 71	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (({𝐴, 𝐵, 𝐶} USGrph 𝐸 ∧ {𝐵, 𝐵} ∈ ran 𝐸) → 𝐵 = 𝐴)
73	72	expcom 450	. . . . . . . . . . . . . . . . 17 ⊢ ({𝐵, 𝐵} ∈ ran 𝐸 → ({𝐴, 𝐵, 𝐶} USGrph 𝐸 → 𝐵 = 𝐴))
74	73	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (({𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐵, 𝐵} ∈ ran 𝐸) → ({𝐴, 𝐵, 𝐶} USGrph 𝐸 → 𝐵 = 𝐴))
75	66, 74	sylbir 224	. . . . . . . . . . . . . . 15 ⊢ ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸 → ({𝐴, 𝐵, 𝐶} USGrph 𝐸 → 𝐵 = 𝐴))
76	75	adantld 482	. . . . . . . . . . . . . 14 ⊢ ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐵 = 𝐴))
77	76	2a1i 12	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐴 → ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐵 = 𝐴))))
78		eqeq2 2621	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝐴 → (𝐵 = 𝑦 ↔ 𝐵 = 𝐴))
79	78	imbi2d 329	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝐴 → (((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐵 = 𝑦) ↔ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐵 = 𝐴)))
80	79	imbi2d 329	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐴 → (({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐵 = 𝑦)) ↔ ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐵 = 𝐴))))
81	77, 11, 80	3imtr4d 282	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐴 → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸 → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐵 = 𝑦))))
82		eqidd 2611	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐵 = 𝐵)
83	82	a1i 11	. . . . . . . . . . . . . 14 ⊢ ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐵 = 𝐵))
84	83	2a1i 12	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐵 → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸 → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐵 = 𝐵))))
85		eqeq2 2621	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝐵 → (𝐵 = 𝑦 ↔ 𝐵 = 𝐵))
86	85	imbi2d 329	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝐵 → (((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐵 = 𝑦) ↔ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐵 = 𝐵)))
87	86	imbi2d 329	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐵 → (({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐵 = 𝑦)) ↔ ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐵 = 𝐵))))
88	84, 33, 87	3imtr4d 282	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐵 → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸 → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐵 = 𝑦))))
89	69	pm2.24i 145	. . . . . . . . . . . . . . . . . . . 20 ⊢ (¬ 𝐵 = 𝐵 → 𝐵 = 𝐶)
90	68, 89	sylbi 206	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵 ≠ 𝐵 → 𝐵 = 𝐶)
91	67, 90	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (({𝐴, 𝐵, 𝐶} USGrph 𝐸 ∧ {𝐵, 𝐵} ∈ ran 𝐸) → 𝐵 = 𝐶)
92	91	expcom 450	. . . . . . . . . . . . . . . . 17 ⊢ ({𝐵, 𝐵} ∈ ran 𝐸 → ({𝐴, 𝐵, 𝐶} USGrph 𝐸 → 𝐵 = 𝐶))
93	92	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (({𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐵, 𝐵} ∈ ran 𝐸) → ({𝐴, 𝐵, 𝐶} USGrph 𝐸 → 𝐵 = 𝐶))
94	66, 93	sylbir 224	. . . . . . . . . . . . . . 15 ⊢ ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸 → ({𝐴, 𝐵, 𝐶} USGrph 𝐸 → 𝐵 = 𝐶))
95	94	adantld 482	. . . . . . . . . . . . . 14 ⊢ ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐵 = 𝐶))
96	95	2a1i 12	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐶 → ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ ran 𝐸 → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐵 = 𝐶))))
97		eqeq2 2621	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝐶 → (𝐵 = 𝑦 ↔ 𝐵 = 𝐶))
98	97	imbi2d 329	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝐶 → (((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐵 = 𝑦) ↔ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐵 = 𝐶)))
99	98	imbi2d 329	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐶 → (({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐵 = 𝑦)) ↔ ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐵 = 𝐶))))
100	96, 49, 99	3imtr4d 282	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐶 → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸 → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐵 = 𝑦))))
101	81, 88, 100	3jaoi 1383	. . . . . . . . . . 11 ⊢ ((𝑦 = 𝐴 ∨ 𝑦 = 𝐵 ∨ 𝑦 = 𝐶) → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸 → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐵 = 𝑦))))
102		preq1 4212	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝐵 → {𝑥, 𝐴} = {𝐵, 𝐴})
103		preq1 4212	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝐵 → {𝑥, 𝐵} = {𝐵, 𝐵})
104	102, 103	preq12d 4220	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝐵 → {{𝑥, 𝐴}, {𝑥, 𝐵}} = {{𝐵, 𝐴}, {𝐵, 𝐵}})
105	104	sseq1d 3595	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐵 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 ↔ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸))
106		eqeq1 2614	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝐵 → (𝑥 = 𝑦 ↔ 𝐵 = 𝑦))
107	106	imbi2d 329	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐵 → (((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝑥 = 𝑦) ↔ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐵 = 𝑦)))
108	105, 107	imbi12d 333	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐵 → (({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝑥 = 𝑦)) ↔ ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐵 = 𝑦))))
109	108	imbi2d 329	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐵 → (({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝑥 = 𝑦))) ↔ ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸 → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐵 = 𝑦)))))
110	101, 109	syl5ibr 235	. . . . . . . . . 10 ⊢ (𝑥 = 𝐵 → ((𝑦 = 𝐴 ∨ 𝑦 = 𝐵 ∨ 𝑦 = 𝐶) → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝑥 = 𝑦)))))
111	21	pm2.24i 145	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (¬ 𝐴 = 𝐴 → 𝐶 = 𝐴)
112	20, 111	sylbi 206	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ≠ 𝐴 → 𝐶 = 𝐴)
113	19, 112	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (({𝐴, 𝐵, 𝐶} USGrph 𝐸 ∧ {𝐴, 𝐴} ∈ ran 𝐸) → 𝐶 = 𝐴)
114	113	expcom 450	. . . . . . . . . . . . . . . . . 18 ⊢ ({𝐴, 𝐴} ∈ ran 𝐸 → ({𝐴, 𝐵, 𝐶} USGrph 𝐸 → 𝐶 = 𝐴))
115	114	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (({𝐴, 𝐴} ∈ ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸) → ({𝐴, 𝐵, 𝐶} USGrph 𝐸 → 𝐶 = 𝐴))
116	18, 115	sylbir 224	. . . . . . . . . . . . . . . 16 ⊢ ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 → ({𝐴, 𝐵, 𝐶} USGrph 𝐸 → 𝐶 = 𝐴))
117	116	adantld 482	. . . . . . . . . . . . . . 15 ⊢ ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐶 = 𝐴))
118	117	a1d 25	. . . . . . . . . . . . . 14 ⊢ ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 → ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐶 = 𝐴)))
119	118	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐴 → ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 → ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐶 = 𝐴))))
120		eqeq2 2621	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝐴 → (𝐶 = 𝑦 ↔ 𝐶 = 𝐴))
121	120	imbi2d 329	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝐴 → (((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐶 = 𝑦) ↔ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐶 = 𝐴)))
122	121	imbi2d 329	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐴 → (({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐶 = 𝑦)) ↔ ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐶 = 𝐴))))
123	119, 11, 122	3imtr4d 282	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐴 → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸 → ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐶 = 𝑦))))
124		pm2.21 119	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (¬ 𝐵 = 𝐵 → (𝐵 = 𝐵 → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → 𝐶 = 𝐵)))
125	68, 124	sylbi 206	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐵 ≠ 𝐵 → (𝐵 = 𝐵 → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → 𝐶 = 𝐵)))
126	67, 69, 125	mpisyl 21	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (({𝐴, 𝐵, 𝐶} USGrph 𝐸 ∧ {𝐵, 𝐵} ∈ ran 𝐸) → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → 𝐶 = 𝐵))
127	126	expcom 450	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ({𝐵, 𝐵} ∈ ran 𝐸 → ({𝐴, 𝐵, 𝐶} USGrph 𝐸 → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → 𝐶 = 𝐵)))
128	127	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ (({𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐵, 𝐵} ∈ ran 𝐸) → ({𝐴, 𝐵, 𝐶} USGrph 𝐸 → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → 𝐶 = 𝐵)))
129	66, 128	sylbir 224	. . . . . . . . . . . . . . . . . . 19 ⊢ ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸 → ({𝐴, 𝐵, 𝐶} USGrph 𝐸 → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → 𝐶 = 𝐵)))
130	129	com13 86	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → ({𝐴, 𝐵, 𝐶} USGrph 𝐸 → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸 → 𝐶 = 𝐵)))
131	130	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → ({𝐴, 𝐵, 𝐶} USGrph 𝐸 → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸 → 𝐶 = 𝐵)))
132	131	imp 444	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸 → 𝐶 = 𝐵))
133	132	com12 32	. . . . . . . . . . . . . . 15 ⊢ ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐶 = 𝐵))
134	133	a1d 25	. . . . . . . . . . . . . 14 ⊢ ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸 → ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐶 = 𝐵)))
135	134	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐵 → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸 → ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐶 = 𝐵))))
136		eqeq2 2621	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝐵 → (𝐶 = 𝑦 ↔ 𝐶 = 𝐵))
137	136	imbi2d 329	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝐵 → (((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐶 = 𝑦) ↔ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐶 = 𝐵)))
138	137	imbi2d 329	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐵 → (({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐶 = 𝑦)) ↔ ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐶 = 𝐵))))
139	135, 33, 138	3imtr4d 282	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐵 → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸 → ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐶 = 𝑦))))
140		eqidd 2611	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐶 = 𝐶)
141	140	a1i 11	. . . . . . . . . . . . . 14 ⊢ ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐶 = 𝐶))
142	141	2a1i 12	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐶 → ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ ran 𝐸 → ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐶 = 𝐶))))
143		eqeq2 2621	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝐶 → (𝐶 = 𝑦 ↔ 𝐶 = 𝐶))
144	143	imbi2d 329	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝐶 → (((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐶 = 𝑦) ↔ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐶 = 𝐶)))
145	144	imbi2d 329	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐶 → (({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐶 = 𝑦)) ↔ ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐶 = 𝐶))))
146	142, 49, 145	3imtr4d 282	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐶 → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸 → ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐶 = 𝑦))))
147	123, 139, 146	3jaoi 1383	. . . . . . . . . . 11 ⊢ ((𝑦 = 𝐴 ∨ 𝑦 = 𝐵 ∨ 𝑦 = 𝐶) → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸 → ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐶 = 𝑦))))
148		preq1 4212	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝐶 → {𝑥, 𝐴} = {𝐶, 𝐴})
149		preq1 4212	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝐶 → {𝑥, 𝐵} = {𝐶, 𝐵})
150	148, 149	preq12d 4220	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝐶 → {{𝑥, 𝐴}, {𝑥, 𝐵}} = {{𝐶, 𝐴}, {𝐶, 𝐵}})
151	150	sseq1d 3595	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐶 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 ↔ {{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ ran 𝐸))
152		eqeq1 2614	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝐶 → (𝑥 = 𝑦 ↔ 𝐶 = 𝑦))
153	152	imbi2d 329	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐶 → (((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝑥 = 𝑦) ↔ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐶 = 𝑦)))
154	151, 153	imbi12d 333	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐶 → (({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝑥 = 𝑦)) ↔ ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐶 = 𝑦))))
155	154	imbi2d 329	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐶 → (({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝑥 = 𝑦))) ↔ ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸 → ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐶 = 𝑦)))))
156	147, 155	syl5ibr 235	. . . . . . . . . 10 ⊢ (𝑥 = 𝐶 → ((𝑦 = 𝐴 ∨ 𝑦 = 𝐵 ∨ 𝑦 = 𝐶) → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝑥 = 𝑦)))))
157	63, 110, 156	3jaoi 1383	. . . . . . . . 9 ⊢ ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ∨ 𝑥 = 𝐶) → ((𝑦 = 𝐴 ∨ 𝑦 = 𝐵 ∨ 𝑦 = 𝐶) → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝑥 = 𝑦)))))
158	157	com3l 87	. . . . . . . 8 ⊢ ((𝑦 = 𝐴 ∨ 𝑦 = 𝐵 ∨ 𝑦 = 𝐶) → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸 → ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ∨ 𝑥 = 𝐶) → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝑥 = 𝑦)))))
159	4, 158	sylbi 206	. . . . . . 7 ⊢ (𝑦 ∈ {𝐴, 𝐵, 𝐶} → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸 → ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ∨ 𝑥 = 𝐶) → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝑥 = 𝑦)))))
160	159	imp 444	. . . . . 6 ⊢ ((𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸) → ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ∨ 𝑥 = 𝐶) → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝑥 = 𝑦))))
161	160	com3l 87	. . . . 5 ⊢ ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ∨ 𝑥 = 𝐶) → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 → ((𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸) → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝑥 = 𝑦))))
162	2, 161	sylbi 206	. . . 4 ⊢ (𝑥 ∈ {𝐴, 𝐵, 𝐶} → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 → ((𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸) → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝑥 = 𝑦))))
163	162	imp31 447	. . 3 ⊢ (((𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸) ∧ (𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸)) → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝑥 = 𝑦))
164	163	com12 32	. 2 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → (((𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸) ∧ (𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸)) → 𝑥 = 𝑦))
165	164	alrimivv 1843	1 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → ∀𝑥∀𝑦(((𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸) ∧ (𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸)) → 𝑥 = 𝑦))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   ∨ w3o 1030   ∧ w3a 1031  ∀wal 1473   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   ⊆ wss 3540  {cpr 4127  {ctp 4129   class class class wbr 4583  ran crn 5039   USGrph cusg 25859

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-hash 12980  df-usgra 25862

This theorem is referenced by:  frgra3vlem2  26528