Theorem 3vfriswmgrlem 28146

Description: Lemma for 3vfriswmgr 28147. (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 977	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → ({𝐴, 𝐴} ∈ 𝐸 ∨ {𝐴, 𝐵} ∈ 𝐸))
2		preq2 4620	. . . . . . . . . 10 ⊢ (𝑤 = 𝐴 → {𝐴, 𝑤} = {𝐴, 𝐴})
3	2	eleq1d 2835	. . . . . . . . 9 ⊢ (𝑤 = 𝐴 → ({𝐴, 𝑤} ∈ 𝐸 ↔ {𝐴, 𝐴} ∈ 𝐸))
4		preq2 4620	. . . . . . . . . 10 ⊢ (𝑤 = 𝐵 → {𝐴, 𝑤} = {𝐴, 𝐵})
5	4	eleq1d 2835	. . . . . . . . 9 ⊢ (𝑤 = 𝐵 → ({𝐴, 𝑤} ∈ 𝐸 ↔ {𝐴, 𝐵} ∈ 𝐸))
6	3, 5	rexprg 4583	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → (∃𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ 𝐸 ↔ ({𝐴, 𝐴} ∈ 𝐸 ∨ {𝐴, 𝐵} ∈ 𝐸)))
7	6	3adant3 1130	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) → (∃𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ 𝐸 ↔ ({𝐴, 𝐴} ∈ 𝐸 ∨ {𝐴, 𝐵} ∈ 𝐸)))
8	7	ad2antrr 726	. . . . . 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 260	. . . . 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 3074	. . . . 5 ⊢ (∃𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ 𝐸 ↔ ∃𝑤(𝑤 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑤} ∈ 𝐸))
11	9, 10	sylib 221	. . . 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 3411	. . . . . . . . 9 ⊢ 𝑤 ∈ V
13	12	elpr 4538	. . . . . . . 8 ⊢ (𝑤 ∈ {𝐴, 𝐵} ↔ (𝑤 = 𝐴 ∨ 𝑤 = 𝐵))
14		vex 3411	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
15	14	elpr 4538	. . . . . . . . . . 11 ⊢ (𝑦 ∈ {𝐴, 𝐵} ↔ (𝑦 = 𝐴 ∨ 𝑦 = 𝐵))
16		eqidd 2760	. . . . . . . . . . . . . . . . . 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 4620	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝐴 → {𝐴, 𝑦} = {𝐴, 𝐴})
20	19	eleq1d 2835	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝐴 → ({𝐴, 𝑦} ∈ 𝐸 ↔ {𝐴, 𝐴} ∈ 𝐸))
21		eqeq2 2771	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝐴 → (𝐴 = 𝑦 ↔ 𝐴 = 𝐴))
22	21	imbi2d 345	. . . . . . . . . . . . . . . . 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 345	. . . . . . . . . . . . . . . 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 298	. . . . . . . . . . . . . . 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 27080	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐺 ∈ USGraph ∧ {𝐴, 𝐴} ∈ 𝐸) → 𝐴 ≠ 𝐴)
27	26	adantll 714	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) ∧ {𝐴, 𝐴} ∈ 𝐸) → 𝐴 ≠ 𝐴)
28		df-ne 2950	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐴 ≠ 𝐴 ↔ ¬ 𝐴 = 𝐴)
29		eqid 2759	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝐴 = 𝐴
30	29	pm2.24i 153	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (¬ 𝐴 = 𝐴 → 𝐴 = 𝐵)
31	28, 30	sylbi 220	. . . . . . . . . . . . . . . . . . . . 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 417	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → ({𝐴, 𝐴} ∈ 𝐸 → 𝐴 = 𝐵))
34	33	ad2antlr 727	. . . . . . . . . . . . . . . . . 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 4620	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝐵 → {𝐴, 𝑦} = {𝐴, 𝐵})
38	37	eleq1d 2835	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝐵 → ({𝐴, 𝑦} ∈ 𝐸 ↔ {𝐴, 𝐵} ∈ 𝐸))
39		eqeq2 2771	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝐵 → (𝐴 = 𝑦 ↔ 𝐴 = 𝐵))
40	39	imbi2d 345	. . . . . . . . . . . . . . . . 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 345	. . . . . . . . . . . . . . . 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 298	. . . . . . . . . . . . . . 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 855	. . . . . . . . . . . . . 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 2763	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝐴 → (𝑤 = 𝑦 ↔ 𝐴 = 𝑦))
45	44	imbi2d 345	. . . . . . . . . . . . . . . 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 349	. . . . . . . . . . . . . . 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 345	. . . . . . . . . . . . . 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 249	. . . . . . . . . . . . 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 220	. . . . . . . . . . . . . . . . . . . . 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 417	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → ({𝐴, 𝐴} ∈ 𝐸 → 𝐵 = 𝐴))
53	52	ad2antlr 727	. . . . . . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝐴 → (𝐵 = 𝑦 ↔ 𝐵 = 𝐴))
57	56	imbi2d 345	. . . . . . . . . . . . . . . . 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 345	. . . . . . . . . . . . . . . 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 298	. . . . . . . . . . . . . . 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 2760	. . . . . . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝐵 → (𝐵 = 𝑦 ↔ 𝐵 = 𝐵))
64	63	imbi2d 345	. . . . . . . . . . . . . . . . 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 345	. . . . . . . . . . . . . . . 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 298	. . . . . . . . . . . . . . 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 855	. . . . . . . . . . . . . 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 2763	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝐵 → (𝑤 = 𝑦 ↔ 𝐵 = 𝑦))
69	68	imbi2d 345	. . . . . . . . . . . . . . . 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 349	. . . . . . . . . . . . . . 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 345	. . . . . . . . . . . . . 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 249	. . . . . . . . . . . . 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 855	. . . . . . . . . . . 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 220	. . . . . . . . . 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 411	. . . . . . . . 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 220	. . . . . . 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 422	. . . . . 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 1930	. . . 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 2833	. . . . . 6 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ {𝐴, 𝐵} ↔ 𝑦 ∈ {𝐴, 𝐵}))
83		preq2 4620	. . . . . . 7 ⊢ (𝑤 = 𝑦 → {𝐴, 𝑤} = {𝐴, 𝑦})
84	83	eleq1d 2835	. . . . . 6 ⊢ (𝑤 = 𝑦 → ({𝐴, 𝑤} ∈ 𝐸 ↔ {𝐴, 𝑦} ∈ 𝐸))
85	82, 84	anbi12d 634	. . . . 5 ⊢ (𝑤 = 𝑦 → ((𝑤 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑤} ∈ 𝐸) ↔ (𝑦 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑦} ∈ 𝐸)))
86	85	eu4 2636	. . . 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 587	. . 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 3075	. . 3 ⊢ (∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ 𝐸 ↔ ∃!𝑤(𝑤 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑤} ∈ 𝐸))
89	87, 88	sylibr 237	. 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 417	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 209   ∧ wa 400   ∨ wo 845   ∧ w3a 1085  ∀wal 1537   = wceq 1539  ∃wex 1782   ∈ wcel 2112  ∃!weu 2588   ≠ wne 2949  ∃wrex 3069  ∃!wreu 3070  {cpr 4517  {ctp 4519  ‘cfv 6328  Vtxcvtx 26873  Edgcedg 26924  USGraphcusgr 27026

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5162  ax-nul 5169  ax-pow 5227  ax-pr 5291  ax-un 7452  ax-cnex 10616  ax-resscn 10617  ax-1cn 10618  ax-icn 10619  ax-addcl 10620  ax-addrcl 10621  ax-mulcl 10622  ax-mulrcl 10623  ax-mulcom 10624  ax-addass 10625  ax-mulass 10626  ax-distr 10627  ax-i2m1 10628  ax-1ne0 10629  ax-1rid 10630  ax-rnegex 10631  ax-rrecex 10632  ax-cnre 10633  ax-pre-lttri 10634  ax-pre-lttrn 10635  ax-pre-ltadd 10636  ax-pre-mulgt0 10637

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2899  df-ne 2950  df-nel 3054  df-ral 3073  df-rex 3074  df-reu 3075  df-rab 3077  df-v 3409  df-sbc 3694  df-csb 3802  df-dif 3857  df-un 3859  df-in 3861  df-ss 3871  df-pss 3873  df-nul 4222  df-if 4414  df-pw 4489  df-sn 4516  df-pr 4518  df-tp 4520  df-op 4522  df-uni 4792  df-int 4832  df-iun 4878  df-br 5026  df-opab 5088  df-mpt 5106  df-tr 5132  df-id 5423  df-eprel 5428  df-po 5436  df-so 5437  df-fr 5476  df-we 5478  df-xp 5523  df-rel 5524  df-cnv 5525  df-co 5526  df-dm 5527  df-rn 5528  df-res 5529  df-ima 5530  df-pred 6119  df-ord 6165  df-on 6166  df-lim 6167  df-suc 6168  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7101  df-ov 7146  df-oprab 7147  df-mpo 7148  df-om 7573  df-1st 7686  df-2nd 7687  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-1o 8105  df-oadd 8109  df-er 8292  df-en 8521  df-dom 8522  df-sdom 8523  df-fin 8524  df-dju 9348  df-card 9386  df-pnf 10700  df-mnf 10701  df-xr 10702  df-ltxr 10703  df-le 10704  df-sub 10895  df-neg 10896  df-nn 11660  df-2 11722  df-n0 11920  df-z 12006  df-uz 12268  df-fz 12925  df-hash 13726  df-edg 26925  df-umgr 26960  df-usgr 27028

This theorem is referenced by:  3vfriswmgr  28147