Theorem 3vfriswmgrlem 30301

Description: Lemma for 3vfriswmgr 30302. (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 980	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → ({𝐴, 𝐴} ∈ 𝐸 ∨ {𝐴, 𝐵} ∈ 𝐸))
2		preq2 4689	. . . . . . . . . 10 ⊢ (𝑤 = 𝐴 → {𝐴, 𝑤} = {𝐴, 𝐴})
3	2	eleq1d 2819	. . . . . . . . 9 ⊢ (𝑤 = 𝐴 → ({𝐴, 𝑤} ∈ 𝐸 ↔ {𝐴, 𝐴} ∈ 𝐸))
4		preq2 4689	. . . . . . . . . 10 ⊢ (𝑤 = 𝐵 → {𝐴, 𝑤} = {𝐴, 𝐵})
5	4	eleq1d 2819	. . . . . . . . 9 ⊢ (𝑤 = 𝐵 → ({𝐴, 𝑤} ∈ 𝐸 ↔ {𝐴, 𝐵} ∈ 𝐸))
6	3, 5	rexprg 4652	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → (∃𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ 𝐸 ↔ ({𝐴, 𝐴} ∈ 𝐸 ∨ {𝐴, 𝐵} ∈ 𝐸)))
7	6	3adant3 1132	. . . . . . 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 257	. . . . 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 3059	. . . . 5 ⊢ (∃𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ 𝐸 ↔ ∃𝑤(𝑤 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑤} ∈ 𝐸))
11	9, 10	sylib 218	. . . 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 3442	. . . . . . . . 9 ⊢ 𝑤 ∈ V
13	12	elpr 4603	. . . . . . . 8 ⊢ (𝑤 ∈ {𝐴, 𝐵} ↔ (𝑤 = 𝐴 ∨ 𝑤 = 𝐵))
14		vex 3442	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
15	14	elpr 4603	. . . . . . . . . . 11 ⊢ (𝑦 ∈ {𝐴, 𝐵} ↔ (𝑦 = 𝐴 ∨ 𝑦 = 𝐵))
16		eqidd 2735	. . . . . . . . . . . . . . . . . 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 4689	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝐴 → {𝐴, 𝑦} = {𝐴, 𝐴})
20	19	eleq1d 2819	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝐴 → ({𝐴, 𝑦} ∈ 𝐸 ↔ {𝐴, 𝐴} ∈ 𝐸))
21		eqeq2 2746	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝐴 → (𝐴 = 𝑦 ↔ 𝐴 = 𝐴))
22	21	imbi2d 340	. . . . . . . . . . . . . . . . 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 340	. . . . . . . . . . . . . . . 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 294	. . . . . . . . . . . . . . 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 29228	. . . . . . . . . . . . . . . . . . . . . 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 2931	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐴 ≠ 𝐴 ↔ ¬ 𝐴 = 𝐴)
29		eqid 2734	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝐴 = 𝐴
30	29	pm2.24i 150	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (¬ 𝐴 = 𝐴 → 𝐴 = 𝐵)
31	28, 30	sylbi 217	. . . . . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . . . 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 4689	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝐵 → {𝐴, 𝑦} = {𝐴, 𝐵})
38	37	eleq1d 2819	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝐵 → ({𝐴, 𝑦} ∈ 𝐸 ↔ {𝐴, 𝐵} ∈ 𝐸))
39		eqeq2 2746	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝐵 → (𝐴 = 𝑦 ↔ 𝐴 = 𝐵))
40	39	imbi2d 340	. . . . . . . . . . . . . . . . 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 340	. . . . . . . . . . . . . . . 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 294	. . . . . . . . . . . . . . 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 857	. . . . . . . . . . . . . 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 2738	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝐴 → (𝑤 = 𝑦 ↔ 𝐴 = 𝑦))
45	44	imbi2d 340	. . . . . . . . . . . . . . . 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 344	. . . . . . . . . . . . . . 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 340	. . . . . . . . . . . . . 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	imbitrrid 246	. . . . . . . . . . . . 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 150	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (¬ 𝐴 = 𝐴 → 𝐵 = 𝐴)
50	28, 49	sylbi 217	. . . . . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . . . 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 2746	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝐴 → (𝐵 = 𝑦 ↔ 𝐵 = 𝐴))
57	56	imbi2d 340	. . . . . . . . . . . . . . . . 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 340	. . . . . . . . . . . . . . . 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 294	. . . . . . . . . . . . . . 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 2735	. . . . . . . . . . . . . . . . . 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 2746	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝐵 → (𝐵 = 𝑦 ↔ 𝐵 = 𝐵))
64	63	imbi2d 340	. . . . . . . . . . . . . . . . 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 340	. . . . . . . . . . . . . . . 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 294	. . . . . . . . . . . . . . 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 857	. . . . . . . . . . . . . 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 2738	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝐵 → (𝑤 = 𝑦 ↔ 𝐵 = 𝑦))
69	68	imbi2d 340	. . . . . . . . . . . . . . . 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 344	. . . . . . . . . . . . . . 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 340	. . . . . . . . . . . . . 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	imbitrrid 246	. . . . . . . . . . . . 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 857	. . . . . . . . . . . 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 217	. . . . . . . . . 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 406	. . . . . . . . 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 217	. . . . . . 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 417	. . . . . 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 2817	. . . . . 6 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ {𝐴, 𝐵} ↔ 𝑦 ∈ {𝐴, 𝐵}))
83		preq2 4689	. . . . . . 7 ⊢ (𝑤 = 𝑦 → {𝐴, 𝑤} = {𝐴, 𝑦})
84	83	eleq1d 2819	. . . . . 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 2613	. . . 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 583	. . 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 3349	. . 3 ⊢ (∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ 𝐸 ↔ ∃!𝑤(𝑤 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑤} ∈ 𝐸))
89	87, 88	sylibr 234	. 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 412	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 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086  ∀wal 1539   = wceq 1541  ∃wex 1780   ∈ wcel 2113  ∃!weu 2566   ≠ wne 2930  ∃wrex 3058  ∃!wreu 3346  {cpr 4580  {ctp 4582  ‘cfv 6490  Vtxcvtx 29018  Edgcedg 29069  USGraphcusgr 29171

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-cnex 11080  ax-resscn 11081  ax-1cn 11082  ax-icn 11083  ax-addcl 11084  ax-addrcl 11085  ax-mulcl 11086  ax-mulrcl 11087  ax-mulcom 11088  ax-addass 11089  ax-mulass 11090  ax-distr 11091  ax-i2m1 11092  ax-1ne0 11093  ax-1rid 11094  ax-rnegex 11095  ax-rrecex 11096  ax-cnre 11097  ax-pre-lttri 11098  ax-pre-lttrn 11099  ax-pre-ltadd 11100  ax-pre-mulgt0 11101

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-int 4901  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-oadd 8399  df-er 8633  df-en 8882  df-dom 8883  df-sdom 8884  df-fin 8885  df-dju 9811  df-card 9849  df-pnf 11166  df-mnf 11167  df-xr 11168  df-ltxr 11169  df-le 11170  df-sub 11364  df-neg 11365  df-nn 12144  df-2 12206  df-n0 12400  df-z 12487  df-uz 12750  df-fz 13422  df-hash 14252  df-edg 29070  df-umgr 29105  df-usgr 29173

This theorem is referenced by:  3vfriswmgr  30302