Theorem 3vfriswmgrlem 30569

Description: Lemma for 3vfriswmgr 30570. (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 994	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → ({𝐴, 𝐴} ∈ 𝐸 ∨ {𝐴, 𝐵} ∈ 𝐸))
2		preq2 4705	. . . . . . . . . 10 ⊢ (𝑤 = 𝐴 → {𝐴, 𝑤} = {𝐴, 𝐴})
3	2	eleq1d 2854	. . . . . . . . 9 ⊢ (𝑤 = 𝐴 → ({𝐴, 𝑤} ∈ 𝐸 ↔ {𝐴, 𝐴} ∈ 𝐸))
4		preq2 4705	. . . . . . . . . 10 ⊢ (𝑤 = 𝐵 → {𝐴, 𝑤} = {𝐴, 𝐵})
5	4	eleq1d 2854	. . . . . . . . 9 ⊢ (𝑤 = 𝐵 → ({𝐴, 𝑤} ∈ 𝐸 ↔ {𝐴, 𝐵} ∈ 𝐸))
6	3, 5	rexprg 4668	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → (∃𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ 𝐸 ↔ ({𝐴, 𝐴} ∈ 𝐸 ∨ {𝐴, 𝐵} ∈ 𝐸)))
7	6	3adant3 1148	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) → (∃𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ 𝐸 ↔ ({𝐴, 𝐴} ∈ 𝐸 ∨ {𝐴, 𝐵} ∈ 𝐸)))
8	7	ad2antrr 738	. . . . . 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 3096	. . . . 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 3467	. . . . . . . . 9 ⊢ 𝑤 ∈ V
13	12	elpr 4619	. . . . . . . 8 ⊢ (𝑤 ∈ {𝐴, 𝐵} ↔ (𝑤 = 𝐴 ∨ 𝑤 = 𝐵))
14		vex 3467	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
15	14	elpr 4619	. . . . . . . . . . 11 ⊢ (𝑦 ∈ {𝐴, 𝐵} ↔ (𝑦 = 𝐴 ∨ 𝑦 = 𝐵))
16		eqidd 2770	. . . . . . . . . . . . . . . . . 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 28	. . . . . . . . . . . . . . . 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 4705	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝐴 → {𝐴, 𝑦} = {𝐴, 𝐴})
20	19	eleq1d 2854	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝐴 → ({𝐴, 𝑦} ∈ 𝐸 ↔ {𝐴, 𝐴} ∈ 𝐸))
21		eqeq2 2781	. . . . . . . . . . . . . . . . . 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 297	. . . . . . . . . . . . . . 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 29497	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐺 ∈ USGraph ∧ {𝐴, 𝐴} ∈ 𝐸) → 𝐴 ≠ 𝐴)
27	26	adantll 726	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) ∧ {𝐴, 𝐴} ∈ 𝐸) → 𝐴 ≠ 𝐴)
28		df-ne 2965	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐴 ≠ 𝐴 ↔ ¬ 𝐴 = 𝐴)
29		eqid 2769	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝐴 = 𝐴
30	29	pm2.24i 151	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (¬ 𝐴 = 𝐴 → 𝐴 = 𝐵)
31	28, 30	sylbi 220	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴 ≠ 𝐴 → 𝐴 = 𝐵)
32	27, 31	syl 18	. . . . . . . . . . . . . . . . . . . 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 739	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → ({𝐴, 𝐴} ∈ 𝐸 → 𝐴 = 𝐵))
35	34	com12 33	. . . . . . . . . . . . . . . . 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 4705	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝐵 → {𝐴, 𝑦} = {𝐴, 𝐵})
38	37	eleq1d 2854	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝐵 → ({𝐴, 𝑦} ∈ 𝐸 ↔ {𝐴, 𝐵} ∈ 𝐸))
39		eqeq2 2781	. . . . . . . . . . . . . . . . . 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 297	. . . . . . . . . . . . . . 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 870	. . . . . . . . . . . . . 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 2773	. . . . . . . . . . . . . . . . 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	imbitrrid 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 151	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (¬ 𝐴 = 𝐴 → 𝐵 = 𝐴)
50	28, 49	sylbi 220	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴 ≠ 𝐴 → 𝐵 = 𝐴)
51	27, 50	syl 18	. . . . . . . . . . . . . . . . . . . 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 739	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → ({𝐴, 𝐴} ∈ 𝐸 → 𝐵 = 𝐴))
54	53	com12 33	. . . . . . . . . . . . . . . . 17 ⊢ ({𝐴, 𝐴} ∈ 𝐸 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → 𝐵 = 𝐴))
55	54	a1i13 28	. . . . . . . . . . . . . . . 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 2781	. . . . . . . . . . . . . . . . . 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 297	. . . . . . . . . . . . . . 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 2770	. . . . . . . . . . . . . . . . . 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 28	. . . . . . . . . . . . . . . 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 2781	. . . . . . . . . . . . . . . . . 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 297	. . . . . . . . . . . . . . 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 870	. . . . . . . . . . . . . 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 2773	. . . . . . . . . . . . . . . . 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	imbitrrid 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 870	. . . . . . . . . . . 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 90	. . . . . . . . . . 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 90	. . . . . . . 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 33	. . . . 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 1955	. . . 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 2852	. . . . . 6 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ {𝐴, 𝐵} ↔ 𝑦 ∈ {𝐴, 𝐵}))
83		preq2 4705	. . . . . . 7 ⊢ (𝑤 = 𝑦 → {𝐴, 𝑤} = {𝐴, 𝑦})
84	83	eleq1d 2854	. . . . . 6 ⊢ (𝑤 = 𝑦 → ({𝐴, 𝑤} ∈ 𝐸 ↔ {𝐴, 𝑦} ∈ 𝐸))
85	82, 84	anbi12d 643	. . . . 5 ⊢ (𝑤 = 𝑦 → ((𝑤 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑤} ∈ 𝐸) ↔ (𝑦 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑦} ∈ 𝐸)))
86	85	eu4 2649	. . . 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 594	. . 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 3377	. . 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 860   ∧ w3a 1101  ∀wal 1565   = wceq 1567  ∃wex 1806   ∈ wcel 2149  ∃!weu 2602   ≠ wne 2964  ∃wrex 3095  ∃!wreu 3374  {cpr 4596  {ctp 4598  ‘cfv 6537  Vtxcvtx 29287  Edgcedg 29338  USGraphcusgr 29440

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11156  ax-resscn 11157  ax-1cn 11158  ax-icn 11159  ax-addcl 11160  ax-addrcl 11161  ax-mulcl 11162  ax-mulrcl 11163  ax-mulcom 11164  ax-addass 11165  ax-mulass 11166  ax-distr 11167  ax-i2m1 11168  ax-1ne0 11169  ax-1rid 11170  ax-rnegex 11171  ax-rrecex 11172  ax-cnre 11173  ax-pre-lttri 11174  ax-pre-lttrn 11175  ax-pre-ltadd 11176  ax-pre-mulgt0 11177

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8453  df-oadd 8457  df-er 8694  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-dju 9887  df-card 9925  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-sub 11443  df-neg 11444  df-nn 12234  df-2 12303  df-n0 12505  df-z 12592  df-uz 12863  df-fz 13536  df-hash 14367  df-edg 29339  df-umgr 29374  df-usgr 29442

This theorem is referenced by:  3vfriswmgr  30570