Theorem 3vfriswmgrlem 28056

Description: Lemma for 3vfriswmgr 28057. (Contributed by Alexander van der Vekens, 6-Oct-2017.) (Revised by AV, 31-Mar-2021.)

Hypotheses

Ref	Expression
3vfriswmgr.v	⊢ 𝑉 = (Vtx‘𝐺)
3vfriswmgr.e	⊢ 𝐸 = (Edg‘𝐺)

Assertion

Ref	Expression
3vfriswmgrlem	⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ({𝐴, 𝐵} ∈ 𝐸 → ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ 𝐸))

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

Proof of Theorem 3vfriswmgrlem

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		animorr 975	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → ({𝐴, 𝐴} ∈ 𝐸 ∨ {𝐴, 𝐵} ∈ 𝐸))
2		preq2 4670	. . . . . . . . . 10 ⊢ (𝑤 = 𝐴 → {𝐴, 𝑤} = {𝐴, 𝐴})
3	2	eleq1d 2897	. . . . . . . . 9 ⊢ (𝑤 = 𝐴 → ({𝐴, 𝑤} ∈ 𝐸 ↔ {𝐴, 𝐴} ∈ 𝐸))
4		preq2 4670	. . . . . . . . . 10 ⊢ (𝑤 = 𝐵 → {𝐴, 𝑤} = {𝐴, 𝐵})
5	4	eleq1d 2897	. . . . . . . . 9 ⊢ (𝑤 = 𝐵 → ({𝐴, 𝑤} ∈ 𝐸 ↔ {𝐴, 𝐵} ∈ 𝐸))
6	3, 5	rexprg 4633	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → (∃𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ 𝐸 ↔ ({𝐴, 𝐴} ∈ 𝐸 ∨ {𝐴, 𝐵} ∈ 𝐸)))
7	6	3adant3 1128	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) → (∃𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ 𝐸 ↔ ({𝐴, 𝐴} ∈ 𝐸 ∨ {𝐴, 𝐵} ∈ 𝐸)))
8	7	ad2antrr 724	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → (∃𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ 𝐸 ↔ ({𝐴, 𝐴} ∈ 𝐸 ∨ {𝐴, 𝐵} ∈ 𝐸)))
9	1, 8	mpbird 259	. . . . 5 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → ∃𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ 𝐸)
10		df-rex 3144	. . . . 5 ⊢ (∃𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ 𝐸 ↔ ∃𝑤(𝑤 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑤} ∈ 𝐸))
11	9, 10	sylib 220	. . . 4 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → ∃𝑤(𝑤 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑤} ∈ 𝐸))
12		vex 3497	. . . . . . . . 9 ⊢ 𝑤 ∈ V
13	12	elpr 4590	. . . . . . . 8 ⊢ (𝑤 ∈ {𝐴, 𝐵} ↔ (𝑤 = 𝐴 ∨ 𝑤 = 𝐵))
14		vex 3497	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
15	14	elpr 4590	. . . . . . . . . . 11 ⊢ (𝑦 ∈ {𝐴, 𝐵} ↔ (𝑦 = 𝐴 ∨ 𝑦 = 𝐵))
16		eqidd 2822	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐴 = 𝐴)
17	16	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ({𝐴, 𝐴} ∈ 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐴 = 𝐴))
18	17	a1i13 27	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝐴 → ({𝐴, 𝐴} ∈ 𝐸 → ({𝐴, 𝐴} ∈ 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐴 = 𝐴))))
19		preq2 4670	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝐴 → {𝐴, 𝑦} = {𝐴, 𝐴})
20	19	eleq1d 2897	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝐴 → ({𝐴, 𝑦} ∈ 𝐸 ↔ {𝐴, 𝐴} ∈ 𝐸))
21		eqeq2 2833	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝐴 → (𝐴 = 𝑦 ↔ 𝐴 = 𝐴))
22	21	imbi2d 343	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝐴 → (((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐴 = 𝑦) ↔ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐴 = 𝐴)))
23	22	imbi2d 343	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝐴 → (({𝐴, 𝐴} ∈ 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐴 = 𝑦)) ↔ ({𝐴, 𝐴} ∈ 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐴 = 𝐴))))
24	18, 20, 23	3imtr4d 296	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝐴 → ({𝐴, 𝑦} ∈ 𝐸 → ({𝐴, 𝐴} ∈ 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐴 = 𝑦))))
25		3vfriswmgr.e	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝐸 = (Edg‘𝐺)
26	25	usgredgne 26988	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐺 ∈ USGraph ∧ {𝐴, 𝐴} ∈ 𝐸) → 𝐴 ≠ 𝐴)
27	26	adantll 712	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) ∧ {𝐴, 𝐴} ∈ 𝐸) → 𝐴 ≠ 𝐴)
28		df-ne 3017	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐴 ≠ 𝐴 ↔ ¬ 𝐴 = 𝐴)
29		eqid 2821	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝐴 = 𝐴
30	29	pm2.24i 153	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (¬ 𝐴 = 𝐴 → 𝐴 = 𝐵)
31	28, 30	sylbi 219	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴 ≠ 𝐴 → 𝐴 = 𝐵)
32	27, 31	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) ∧ {𝐴, 𝐴} ∈ 𝐸) → 𝐴 = 𝐵)
33	32	ex 415	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → ({𝐴, 𝐴} ∈ 𝐸 → 𝐴 = 𝐵))
34	33	ad2antlr 725	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → ({𝐴, 𝐴} ∈ 𝐸 → 𝐴 = 𝐵))
35	34	com12 32	. . . . . . . . . . . . . . . . 17 ⊢ ({𝐴, 𝐴} ∈ 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐴 = 𝐵))
36	35	2a1i 12	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝐵 → ({𝐴, 𝐵} ∈ 𝐸 → ({𝐴, 𝐴} ∈ 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐴 = 𝐵))))
37		preq2 4670	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝐵 → {𝐴, 𝑦} = {𝐴, 𝐵})
38	37	eleq1d 2897	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝐵 → ({𝐴, 𝑦} ∈ 𝐸 ↔ {𝐴, 𝐵} ∈ 𝐸))
39		eqeq2 2833	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝐵 → (𝐴 = 𝑦 ↔ 𝐴 = 𝐵))
40	39	imbi2d 343	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝐵 → (((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐴 = 𝑦) ↔ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐴 = 𝐵)))
41	40	imbi2d 343	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝐵 → (({𝐴, 𝐴} ∈ 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐴 = 𝑦)) ↔ ({𝐴, 𝐴} ∈ 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐴 = 𝐵))))
42	36, 38, 41	3imtr4d 296	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝐵 → ({𝐴, 𝑦} ∈ 𝐸 → ({𝐴, 𝐴} ∈ 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐴 = 𝑦))))
43	24, 42	jaoi 853	. . . . . . . . . . . . . 14 ⊢ ((𝑦 = 𝐴 ∨ 𝑦 = 𝐵) → ({𝐴, 𝑦} ∈ 𝐸 → ({𝐴, 𝐴} ∈ 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐴 = 𝑦))))
44		eqeq1 2825	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝐴 → (𝑤 = 𝑦 ↔ 𝐴 = 𝑦))
45	44	imbi2d 343	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝐴 → (((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝑤 = 𝑦) ↔ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐴 = 𝑦)))
46	3, 45	imbi12d 347	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝐴 → (({𝐴, 𝑤} ∈ 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝑤 = 𝑦)) ↔ ({𝐴, 𝐴} ∈ 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐴 = 𝑦))))
47	46	imbi2d 343	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝐴 → (({𝐴, 𝑦} ∈ 𝐸 → ({𝐴, 𝑤} ∈ 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝑤 = 𝑦))) ↔ ({𝐴, 𝑦} ∈ 𝐸 → ({𝐴, 𝐴} ∈ 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐴 = 𝑦)))))
48	43, 47	syl5ibr 248	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝐴 → ((𝑦 = 𝐴 ∨ 𝑦 = 𝐵) → ({𝐴, 𝑦} ∈ 𝐸 → ({𝐴, 𝑤} ∈ 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝑤 = 𝑦)))))
49	29	pm2.24i 153	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (¬ 𝐴 = 𝐴 → 𝐵 = 𝐴)
50	28, 49	sylbi 219	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴 ≠ 𝐴 → 𝐵 = 𝐴)
51	27, 50	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) ∧ {𝐴, 𝐴} ∈ 𝐸) → 𝐵 = 𝐴)
52	51	ex 415	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → ({𝐴, 𝐴} ∈ 𝐸 → 𝐵 = 𝐴))
53	52	ad2antlr 725	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → ({𝐴, 𝐴} ∈ 𝐸 → 𝐵 = 𝐴))
54	53	com12 32	. . . . . . . . . . . . . . . . 17 ⊢ ({𝐴, 𝐴} ∈ 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐵 = 𝐴))
55	54	a1i13 27	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝐴 → ({𝐴, 𝐴} ∈ 𝐸 → ({𝐴, 𝐵} ∈ 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐵 = 𝐴))))
56		eqeq2 2833	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝐴 → (𝐵 = 𝑦 ↔ 𝐵 = 𝐴))
57	56	imbi2d 343	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝐴 → (((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐵 = 𝑦) ↔ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐵 = 𝐴)))
58	57	imbi2d 343	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝐴 → (({𝐴, 𝐵} ∈ 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐵 = 𝑦)) ↔ ({𝐴, 𝐵} ∈ 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐵 = 𝐴))))
59	55, 20, 58	3imtr4d 296	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝐴 → ({𝐴, 𝑦} ∈ 𝐸 → ({𝐴, 𝐵} ∈ 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐵 = 𝑦))))
60		eqidd 2822	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐵 = 𝐵)
61	60	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ({𝐴, 𝐵} ∈ 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐵 = 𝐵))
62	61	a1i13 27	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝐵 → ({𝐴, 𝐵} ∈ 𝐸 → ({𝐴, 𝐵} ∈ 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐵 = 𝐵))))
63		eqeq2 2833	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝐵 → (𝐵 = 𝑦 ↔ 𝐵 = 𝐵))
64	63	imbi2d 343	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝐵 → (((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐵 = 𝑦) ↔ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐵 = 𝐵)))
65	64	imbi2d 343	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝐵 → (({𝐴, 𝐵} ∈ 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐵 = 𝑦)) ↔ ({𝐴, 𝐵} ∈ 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐵 = 𝐵))))
66	62, 38, 65	3imtr4d 296	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝐵 → ({𝐴, 𝑦} ∈ 𝐸 → ({𝐴, 𝐵} ∈ 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐵 = 𝑦))))
67	59, 66	jaoi 853	. . . . . . . . . . . . . 14 ⊢ ((𝑦 = 𝐴 ∨ 𝑦 = 𝐵) → ({𝐴, 𝑦} ∈ 𝐸 → ({𝐴, 𝐵} ∈ 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐵 = 𝑦))))
68		eqeq1 2825	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝐵 → (𝑤 = 𝑦 ↔ 𝐵 = 𝑦))
69	68	imbi2d 343	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝐵 → (((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝑤 = 𝑦) ↔ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐵 = 𝑦)))
70	5, 69	imbi12d 347	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝐵 → (({𝐴, 𝑤} ∈ 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝑤 = 𝑦)) ↔ ({𝐴, 𝐵} ∈ 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐵 = 𝑦))))
71	70	imbi2d 343	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝐵 → (({𝐴, 𝑦} ∈ 𝐸 → ({𝐴, 𝑤} ∈ 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝑤 = 𝑦))) ↔ ({𝐴, 𝑦} ∈ 𝐸 → ({𝐴, 𝐵} ∈ 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐵 = 𝑦)))))
72	67, 71	syl5ibr 248	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝐵 → ((𝑦 = 𝐴 ∨ 𝑦 = 𝐵) → ({𝐴, 𝑦} ∈ 𝐸 → ({𝐴, 𝑤} ∈ 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝑤 = 𝑦)))))
73	48, 72	jaoi 853	. . . . . . . . . . . 12 ⊢ ((𝑤 = 𝐴 ∨ 𝑤 = 𝐵) → ((𝑦 = 𝐴 ∨ 𝑦 = 𝐵) → ({𝐴, 𝑦} ∈ 𝐸 → ({𝐴, 𝑤} ∈ 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝑤 = 𝑦)))))
74	73	com3l 89	. . . . . . . . . . 11 ⊢ ((𝑦 = 𝐴 ∨ 𝑦 = 𝐵) → ({𝐴, 𝑦} ∈ 𝐸 → ((𝑤 = 𝐴 ∨ 𝑤 = 𝐵) → ({𝐴, 𝑤} ∈ 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝑤 = 𝑦)))))
75	15, 74	sylbi 219	. . . . . . . . . 10 ⊢ (𝑦 ∈ {𝐴, 𝐵} → ({𝐴, 𝑦} ∈ 𝐸 → ((𝑤 = 𝐴 ∨ 𝑤 = 𝐵) → ({𝐴, 𝑤} ∈ 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝑤 = 𝑦)))))
76	75	imp 409	. . . . . . . . 9 ⊢ ((𝑦 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑦} ∈ 𝐸) → ((𝑤 = 𝐴 ∨ 𝑤 = 𝐵) → ({𝐴, 𝑤} ∈ 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝑤 = 𝑦))))
77	76	com3l 89	. . . . . . . 8 ⊢ ((𝑤 = 𝐴 ∨ 𝑤 = 𝐵) → ({𝐴, 𝑤} ∈ 𝐸 → ((𝑦 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑦} ∈ 𝐸) → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝑤 = 𝑦))))
78	13, 77	sylbi 219	. . . . . . 7 ⊢ (𝑤 ∈ {𝐴, 𝐵} → ({𝐴, 𝑤} ∈ 𝐸 → ((𝑦 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑦} ∈ 𝐸) → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝑤 = 𝑦))))
79	78	imp31 420	. . . . . 6 ⊢ (((𝑤 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑤} ∈ 𝐸) ∧ (𝑦 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑦} ∈ 𝐸)) → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝑤 = 𝑦))
80	79	com12 32	. . . . 5 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → (((𝑤 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑤} ∈ 𝐸) ∧ (𝑦 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑦} ∈ 𝐸)) → 𝑤 = 𝑦))
81	80	alrimivv 1929	. . . 4 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → ∀𝑤∀𝑦(((𝑤 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑤} ∈ 𝐸) ∧ (𝑦 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑦} ∈ 𝐸)) → 𝑤 = 𝑦))
82		eleq1w 2895	. . . . . 6 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ {𝐴, 𝐵} ↔ 𝑦 ∈ {𝐴, 𝐵}))
83		preq2 4670	. . . . . . 7 ⊢ (𝑤 = 𝑦 → {𝐴, 𝑤} = {𝐴, 𝑦})
84	83	eleq1d 2897	. . . . . 6 ⊢ (𝑤 = 𝑦 → ({𝐴, 𝑤} ∈ 𝐸 ↔ {𝐴, 𝑦} ∈ 𝐸))
85	82, 84	anbi12d 632	. . . . 5 ⊢ (𝑤 = 𝑦 → ((𝑤 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑤} ∈ 𝐸) ↔ (𝑦 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑦} ∈ 𝐸)))
86	85	eu4 2699	. . . 4 ⊢ (∃!𝑤(𝑤 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑤} ∈ 𝐸) ↔ (∃𝑤(𝑤 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑤} ∈ 𝐸) ∧ ∀𝑤∀𝑦(((𝑤 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑤} ∈ 𝐸) ∧ (𝑦 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑦} ∈ 𝐸)) → 𝑤 = 𝑦)))
87	11, 81, 86	sylanbrc 585	. . 3 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → ∃!𝑤(𝑤 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑤} ∈ 𝐸))
88		df-reu 3145	. . 3 ⊢ (∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ 𝐸 ↔ ∃!𝑤(𝑤 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑤} ∈ 𝐸))
89	87, 88	sylibr 236	. 2 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ 𝐸)
90	89	ex 415	1 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ({𝐴, 𝐵} ∈ 𝐸 → ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ 𝐸))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843   ∧ w3a 1083  ∀wal 1535   = wceq 1537  ∃wex 1780   ∈ wcel 2114  ∃!weu 2653   ≠ wne 3016  ∃wrex 3139  ∃!wreu 3140  {cpr 4569  {ctp 4571  ‘cfv 6355  Vtxcvtx 26781  Edgcedg 26832  USGraphcusgr 26934

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-1o 8102  df-oadd 8106  df-er 8289  df-en 8510  df-dom 8511  df-sdom 8512  df-fin 8513  df-dju 9330  df-card 9368  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-nn 11639  df-2 11701  df-n0 11899  df-z 11983  df-uz 12245  df-fz 12894  df-hash 13692  df-edg 26833  df-umgr 26868  df-usgr 26936

This theorem is referenced by:  3vfriswmgr  28057