Theorem uhgr2edg 16188

Description: If a vertex is adjacent to two different vertices in a hypergraph, there are more than one edges starting at this vertex. (Contributed by Alexander van der Vekens, 10-Dec-2017.) (Revised by AV, 11-Feb-2021.)

Hypotheses

Ref	Expression
usgrf1oedg.i	⊢ 𝐼 = (iEdg‘𝐺)
usgrf1oedg.e	⊢ 𝐸 = (Edg‘𝐺)
uhgr2edg.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
uhgr2edg	⊢ (((𝐺 ∈ UHGraph ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ∧ ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸)) → ∃𝑥 ∈ dom 𝐼∃𝑦 ∈ dom 𝐼(𝑥 ≠ 𝑦 ∧ 𝑁 ∈ (𝐼‘𝑥) ∧ 𝑁 ∈ (𝐼‘𝑦)))

Distinct variable groups:   𝑥,𝐺   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑦,𝐺   𝑥,𝐼,𝑦   𝑥,𝑁,𝑦   𝑥,𝑉,𝑦

Allowed substitution hints:   𝐸(𝑥,𝑦)

Proof of Theorem uhgr2edg

Step	Hyp	Ref	Expression
1		simp1l 1048	. . 3 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ∧ ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸)) → 𝐺 ∈ UHGraph)
2		simp1r 1049	. . 3 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ∧ ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸)) → 𝐴 ≠ 𝐵)
3		simp23 1059	. . . 4 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ∧ ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸)) → 𝑁 ∈ 𝑉)
4		simp21 1057	. . . 4 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ∧ ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸)) → 𝐴 ∈ 𝑉)
5		3simpc 1023	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) → (𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))
6	5	3ad2ant2 1046	. . . 4 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ∧ ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸)) → (𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))
7	3, 4, 6	jca31 309	. . 3 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ∧ ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸)) → ((𝑁 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉)))
8	1, 2, 7	jca31 309	. 2 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ∧ ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸)) → ((𝐺 ∈ UHGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑁 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))))
9		simp3 1026	. 2 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ∧ ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸)) → ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸))
10		usgrf1oedg.e	. . . . . . . . 9 ⊢ 𝐸 = (Edg‘𝐺)
11	10	a1i 9	. . . . . . . 8 ⊢ (𝐺 ∈ UHGraph → 𝐸 = (Edg‘𝐺))
12		edgvalg 16041	. . . . . . . 8 ⊢ (𝐺 ∈ UHGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
13		usgrf1oedg.i	. . . . . . . . . . 11 ⊢ 𝐼 = (iEdg‘𝐺)
14	13	eqcomi 2236	. . . . . . . . . 10 ⊢ (iEdg‘𝐺) = 𝐼
15	14	a1i 9	. . . . . . . . 9 ⊢ (𝐺 ∈ UHGraph → (iEdg‘𝐺) = 𝐼)
16	15	rneqd 4985	. . . . . . . 8 ⊢ (𝐺 ∈ UHGraph → ran (iEdg‘𝐺) = ran 𝐼)
17	11, 12, 16	3eqtrd 2269	. . . . . . 7 ⊢ (𝐺 ∈ UHGraph → 𝐸 = ran 𝐼)
18	17	eleq2d 2302	. . . . . 6 ⊢ (𝐺 ∈ UHGraph → ({𝑁, 𝐴} ∈ 𝐸 ↔ {𝑁, 𝐴} ∈ ran 𝐼))
19	17	eleq2d 2302	. . . . . 6 ⊢ (𝐺 ∈ UHGraph → ({𝐵, 𝑁} ∈ 𝐸 ↔ {𝐵, 𝑁} ∈ ran 𝐼))
20	18, 19	anbi12d 473	. . . . 5 ⊢ (𝐺 ∈ UHGraph → (({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸) ↔ ({𝑁, 𝐴} ∈ ran 𝐼 ∧ {𝐵, 𝑁} ∈ ran 𝐼)))
21	13	uhgrfun 16059	. . . . . . 7 ⊢ (𝐺 ∈ UHGraph → Fun 𝐼)
22	21	funfnd 5382	. . . . . 6 ⊢ (𝐺 ∈ UHGraph → 𝐼 Fn dom 𝐼)
23		fvelrnb 5723	. . . . . . 7 ⊢ (𝐼 Fn dom 𝐼 → ({𝑁, 𝐴} ∈ ran 𝐼 ↔ ∃𝑥 ∈ dom 𝐼(𝐼‘𝑥) = {𝑁, 𝐴}))
24		fvelrnb 5723	. . . . . . 7 ⊢ (𝐼 Fn dom 𝐼 → ({𝐵, 𝑁} ∈ ran 𝐼 ↔ ∃𝑦 ∈ dom 𝐼(𝐼‘𝑦) = {𝐵, 𝑁}))
25	23, 24	anbi12d 473	. . . . . 6 ⊢ (𝐼 Fn dom 𝐼 → (({𝑁, 𝐴} ∈ ran 𝐼 ∧ {𝐵, 𝑁} ∈ ran 𝐼) ↔ (∃𝑥 ∈ dom 𝐼(𝐼‘𝑥) = {𝑁, 𝐴} ∧ ∃𝑦 ∈ dom 𝐼(𝐼‘𝑦) = {𝐵, 𝑁})))
26	22, 25	syl 14	. . . . 5 ⊢ (𝐺 ∈ UHGraph → (({𝑁, 𝐴} ∈ ran 𝐼 ∧ {𝐵, 𝑁} ∈ ran 𝐼) ↔ (∃𝑥 ∈ dom 𝐼(𝐼‘𝑥) = {𝑁, 𝐴} ∧ ∃𝑦 ∈ dom 𝐼(𝐼‘𝑦) = {𝐵, 𝑁})))
27	20, 26	bitrd 188	. . . 4 ⊢ (𝐺 ∈ UHGraph → (({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸) ↔ (∃𝑥 ∈ dom 𝐼(𝐼‘𝑥) = {𝑁, 𝐴} ∧ ∃𝑦 ∈ dom 𝐼(𝐼‘𝑦) = {𝐵, 𝑁})))
28	27	ad2antrr 488	. . 3 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑁 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))) → (({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸) ↔ (∃𝑥 ∈ dom 𝐼(𝐼‘𝑥) = {𝑁, 𝐴} ∧ ∃𝑦 ∈ dom 𝐼(𝐼‘𝑦) = {𝐵, 𝑁})))
29		reeanv 2713	. . . 4 ⊢ (∃𝑥 ∈ dom 𝐼∃𝑦 ∈ dom 𝐼((𝐼‘𝑥) = {𝑁, 𝐴} ∧ (𝐼‘𝑦) = {𝐵, 𝑁}) ↔ (∃𝑥 ∈ dom 𝐼(𝐼‘𝑥) = {𝑁, 𝐴} ∧ ∃𝑦 ∈ dom 𝐼(𝐼‘𝑦) = {𝐵, 𝑁}))
30		fveqeq2 5678	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → ((𝐼‘𝑥) = {𝑁, 𝐴} ↔ (𝐼‘𝑦) = {𝑁, 𝐴}))
31	30	anbi1d 465	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → (((𝐼‘𝑥) = {𝑁, 𝐴} ∧ (𝐼‘𝑦) = {𝐵, 𝑁}) ↔ ((𝐼‘𝑦) = {𝑁, 𝐴} ∧ (𝐼‘𝑦) = {𝐵, 𝑁})))
32		eqtr2 2251	. . . . . . . . . . . . . . 15 ⊢ (((𝐼‘𝑦) = {𝑁, 𝐴} ∧ (𝐼‘𝑦) = {𝐵, 𝑁}) → {𝑁, 𝐴} = {𝐵, 𝑁})
33		prcom 3766	. . . . . . . . . . . . . . . . 17 ⊢ {𝐵, 𝑁} = {𝑁, 𝐵}
34	33	eqeq2i 2243	. . . . . . . . . . . . . . . 16 ⊢ ({𝑁, 𝐴} = {𝐵, 𝑁} ↔ {𝑁, 𝐴} = {𝑁, 𝐵})
35		preq12bg 3876	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑁 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ (𝑁 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → ({𝑁, 𝐴} = {𝑁, 𝐵} ↔ ((𝑁 = 𝑁 ∧ 𝐴 = 𝐵) ∨ (𝑁 = 𝐵 ∧ 𝐴 = 𝑁))))
36	35	ancom2s 568	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉)) → ({𝑁, 𝐴} = {𝑁, 𝐵} ↔ ((𝑁 = 𝑁 ∧ 𝐴 = 𝐵) ∨ (𝑁 = 𝐵 ∧ 𝐴 = 𝑁))))
37		eqneqall 2422	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → 𝑥 ≠ 𝑦))
38	37	adantl 277	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑁 = 𝑁 ∧ 𝐴 = 𝐵) → (𝐴 ≠ 𝐵 → 𝑥 ≠ 𝑦))
39		eqtr 2250	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 = 𝑁 ∧ 𝑁 = 𝐵) → 𝐴 = 𝐵)
40	39	ancoms 268	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑁 = 𝐵 ∧ 𝐴 = 𝑁) → 𝐴 = 𝐵)
41	40, 37	syl 14	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑁 = 𝐵 ∧ 𝐴 = 𝑁) → (𝐴 ≠ 𝐵 → 𝑥 ≠ 𝑦))
42	38, 41	jaoi 724	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑁 = 𝑁 ∧ 𝐴 = 𝐵) ∨ (𝑁 = 𝐵 ∧ 𝐴 = 𝑁)) → (𝐴 ≠ 𝐵 → 𝑥 ≠ 𝑦))
43	42	adantld 278	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 = 𝑁 ∧ 𝐴 = 𝐵) ∨ (𝑁 = 𝐵 ∧ 𝐴 = 𝑁)) → ((𝐺 ∈ UHGraph ∧ 𝐴 ≠ 𝐵) → 𝑥 ≠ 𝑦))
44	36, 43	biimtrdi 163	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉)) → ({𝑁, 𝐴} = {𝑁, 𝐵} → ((𝐺 ∈ UHGraph ∧ 𝐴 ≠ 𝐵) → 𝑥 ≠ 𝑦)))
45	44	com3l 81	. . . . . . . . . . . . . . . . 17 ⊢ ({𝑁, 𝐴} = {𝑁, 𝐵} → ((𝐺 ∈ UHGraph ∧ 𝐴 ≠ 𝐵) → (((𝑁 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉)) → 𝑥 ≠ 𝑦)))
46	45	impd 254	. . . . . . . . . . . . . . . 16 ⊢ ({𝑁, 𝐴} = {𝑁, 𝐵} → (((𝐺 ∈ UHGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑁 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))) → 𝑥 ≠ 𝑦))
47	34, 46	sylbi 121	. . . . . . . . . . . . . . 15 ⊢ ({𝑁, 𝐴} = {𝐵, 𝑁} → (((𝐺 ∈ UHGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑁 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))) → 𝑥 ≠ 𝑦))
48	32, 47	syl 14	. . . . . . . . . . . . . 14 ⊢ (((𝐼‘𝑦) = {𝑁, 𝐴} ∧ (𝐼‘𝑦) = {𝐵, 𝑁}) → (((𝐺 ∈ UHGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑁 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))) → 𝑥 ≠ 𝑦))
49	31, 48	biimtrdi 163	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (((𝐼‘𝑥) = {𝑁, 𝐴} ∧ (𝐼‘𝑦) = {𝐵, 𝑁}) → (((𝐺 ∈ UHGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑁 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))) → 𝑥 ≠ 𝑦)))
50	49	impcomd 255	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → ((((𝐺 ∈ UHGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑁 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))) ∧ ((𝐼‘𝑥) = {𝑁, 𝐴} ∧ (𝐼‘𝑦) = {𝐵, 𝑁})) → 𝑥 ≠ 𝑦))
51	50	impcom 125	. . . . . . . . . . 11 ⊢ (((((𝐺 ∈ UHGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑁 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))) ∧ ((𝐼‘𝑥) = {𝑁, 𝐴} ∧ (𝐼‘𝑦) = {𝐵, 𝑁})) ∧ 𝑥 = 𝑦) → 𝑥 ≠ 𝑦)
52	51	neneqd 2433	. . . . . . . . . 10 ⊢ (((((𝐺 ∈ UHGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑁 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))) ∧ ((𝐼‘𝑥) = {𝑁, 𝐴} ∧ (𝐼‘𝑦) = {𝐵, 𝑁})) ∧ 𝑥 = 𝑦) → ¬ 𝑥 = 𝑦)
53	52	pm2.01da 641	. . . . . . . . 9 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑁 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))) ∧ ((𝐼‘𝑥) = {𝑁, 𝐴} ∧ (𝐼‘𝑦) = {𝐵, 𝑁})) → ¬ 𝑥 = 𝑦)
54	53	neqned 2419	. . . . . . . 8 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑁 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))) ∧ ((𝐼‘𝑥) = {𝑁, 𝐴} ∧ (𝐼‘𝑦) = {𝐵, 𝑁})) → 𝑥 ≠ 𝑦)
55		prid1g 3794	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ 𝑉 → 𝑁 ∈ {𝑁, 𝐴})
56	55	ad2antrr 488	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉)) → 𝑁 ∈ {𝑁, 𝐴})
57	56	adantl 277	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑁 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))) → 𝑁 ∈ {𝑁, 𝐴})
58		eleq2 2296	. . . . . . . . . . 11 ⊢ ((𝐼‘𝑥) = {𝑁, 𝐴} → (𝑁 ∈ (𝐼‘𝑥) ↔ 𝑁 ∈ {𝑁, 𝐴}))
59	57, 58	imbitrrid 156	. . . . . . . . . 10 ⊢ ((𝐼‘𝑥) = {𝑁, 𝐴} → (((𝐺 ∈ UHGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑁 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))) → 𝑁 ∈ (𝐼‘𝑥)))
60	59	adantr 276	. . . . . . . . 9 ⊢ (((𝐼‘𝑥) = {𝑁, 𝐴} ∧ (𝐼‘𝑦) = {𝐵, 𝑁}) → (((𝐺 ∈ UHGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑁 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))) → 𝑁 ∈ (𝐼‘𝑥)))
61	60	impcom 125	. . . . . . . 8 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑁 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))) ∧ ((𝐼‘𝑥) = {𝑁, 𝐴} ∧ (𝐼‘𝑦) = {𝐵, 𝑁})) → 𝑁 ∈ (𝐼‘𝑥))
62		prid2g 3795	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ 𝑉 → 𝑁 ∈ {𝐵, 𝑁})
63	62	ad2antrr 488	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉)) → 𝑁 ∈ {𝐵, 𝑁})
64	63	adantl 277	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑁 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))) → 𝑁 ∈ {𝐵, 𝑁})
65		eleq2 2296	. . . . . . . . . . 11 ⊢ ((𝐼‘𝑦) = {𝐵, 𝑁} → (𝑁 ∈ (𝐼‘𝑦) ↔ 𝑁 ∈ {𝐵, 𝑁}))
66	64, 65	imbitrrid 156	. . . . . . . . . 10 ⊢ ((𝐼‘𝑦) = {𝐵, 𝑁} → (((𝐺 ∈ UHGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑁 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))) → 𝑁 ∈ (𝐼‘𝑦)))
67	66	adantl 277	. . . . . . . . 9 ⊢ (((𝐼‘𝑥) = {𝑁, 𝐴} ∧ (𝐼‘𝑦) = {𝐵, 𝑁}) → (((𝐺 ∈ UHGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑁 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))) → 𝑁 ∈ (𝐼‘𝑦)))
68	67	impcom 125	. . . . . . . 8 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑁 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))) ∧ ((𝐼‘𝑥) = {𝑁, 𝐴} ∧ (𝐼‘𝑦) = {𝐵, 𝑁})) → 𝑁 ∈ (𝐼‘𝑦))
69	54, 61, 68	3jca 1204	. . . . . . 7 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑁 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))) ∧ ((𝐼‘𝑥) = {𝑁, 𝐴} ∧ (𝐼‘𝑦) = {𝐵, 𝑁})) → (𝑥 ≠ 𝑦 ∧ 𝑁 ∈ (𝐼‘𝑥) ∧ 𝑁 ∈ (𝐼‘𝑦)))
70	69	ex 115	. . . . . 6 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑁 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))) → (((𝐼‘𝑥) = {𝑁, 𝐴} ∧ (𝐼‘𝑦) = {𝐵, 𝑁}) → (𝑥 ≠ 𝑦 ∧ 𝑁 ∈ (𝐼‘𝑥) ∧ 𝑁 ∈ (𝐼‘𝑦))))
71	70	reximdv 2643	. . . . 5 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑁 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))) → (∃𝑦 ∈ dom 𝐼((𝐼‘𝑥) = {𝑁, 𝐴} ∧ (𝐼‘𝑦) = {𝐵, 𝑁}) → ∃𝑦 ∈ dom 𝐼(𝑥 ≠ 𝑦 ∧ 𝑁 ∈ (𝐼‘𝑥) ∧ 𝑁 ∈ (𝐼‘𝑦))))
72	71	reximdv 2643	. . . 4 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑁 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))) → (∃𝑥 ∈ dom 𝐼∃𝑦 ∈ dom 𝐼((𝐼‘𝑥) = {𝑁, 𝐴} ∧ (𝐼‘𝑦) = {𝐵, 𝑁}) → ∃𝑥 ∈ dom 𝐼∃𝑦 ∈ dom 𝐼(𝑥 ≠ 𝑦 ∧ 𝑁 ∈ (𝐼‘𝑥) ∧ 𝑁 ∈ (𝐼‘𝑦))))
73	29, 72	biimtrrid 153	. . 3 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑁 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))) → ((∃𝑥 ∈ dom 𝐼(𝐼‘𝑥) = {𝑁, 𝐴} ∧ ∃𝑦 ∈ dom 𝐼(𝐼‘𝑦) = {𝐵, 𝑁}) → ∃𝑥 ∈ dom 𝐼∃𝑦 ∈ dom 𝐼(𝑥 ≠ 𝑦 ∧ 𝑁 ∈ (𝐼‘𝑥) ∧ 𝑁 ∈ (𝐼‘𝑦))))
74	28, 73	sylbid 150	. 2 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑁 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))) → (({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸) → ∃𝑥 ∈ dom 𝐼∃𝑦 ∈ dom 𝐼(𝑥 ≠ 𝑦 ∧ 𝑁 ∈ (𝐼‘𝑥) ∧ 𝑁 ∈ (𝐼‘𝑦))))
75	8, 9, 74	sylc 62	1 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ∧ ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸)) → ∃𝑥 ∈ dom 𝐼∃𝑦 ∈ dom 𝐼(𝑥 ≠ 𝑦 ∧ 𝑁 ∈ (𝐼‘𝑥) ∧ 𝑁 ∈ (𝐼‘𝑦)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 716   ∧ w3a 1005   = wceq 1398   ∈ wcel 2203   ≠ wne 2412  ∃wrex 2521  {cpr 3689  dom cdm 4748  ran crn 4749   Fn wfn 5346  ‘cfv 5351  Vtxcvtx 15994  iEdgciedg 15995  Edgcedg 16039  UHGraphcuhgr 16049

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-cnex 8214  ax-resscn 8215  ax-1cn 8216  ax-1re 8217  ax-icn 8218  ax-addcl 8219  ax-addrcl 8220  ax-mulcl 8221  ax-addcom 8223  ax-mulcom 8224  ax-addass 8225  ax-mulass 8226  ax-distr 8227  ax-i2m1 8228  ax-1rid 8230  ax-0id 8231  ax-rnegex 8232  ax-cnre 8234

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-fo 5357  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-sub 8442  df-inn 9234  df-2 9292  df-3 9293  df-4 9294  df-5 9295  df-6 9296  df-7 9297  df-8 9298  df-9 9299  df-n0 9493  df-dec 9706  df-ndx 13204  df-slot 13205  df-base 13207  df-edgf 15987  df-vtx 15996  df-iedg 15997  df-edg 16040  df-uhgrm 16051

This theorem is referenced by:  umgr2edg  16189