Theorem uhgr2edg 40435

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 1078	. . . 4 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ∧ ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸)) → 𝐺 ∈ UHGraph )
2		simp1r 1079	. . . 4 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ∧ ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸)) → 𝐴 ≠ 𝐵)
3		simp23 1089	. . . . 5 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ∧ ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸)) → 𝑁 ∈ 𝑉)
4		simp21 1087	. . . . 5 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ∧ ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸)) → 𝐴 ∈ 𝑉)
5		3simpc 1053	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) → (𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))
6	5	3ad2ant2 1076	. . . . 5 ⊢ (((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 555	. . . 4 ⊢ (((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 555	. . 3 ⊢ (((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 1056	. . 3 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ∧ ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸)) → ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸))
10	8, 9	jca 553	. 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))) ∧ ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸)))
11		usgrf1oedg.e	. . . . . . . . . 10 ⊢ 𝐸 = (Edg‘𝐺)
12	11	a1i 11	. . . . . . . . 9 ⊢ (𝐺 ∈ UHGraph → 𝐸 = (Edg‘𝐺))
13		edgaval 25794	. . . . . . . . 9 ⊢ (𝐺 ∈ UHGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
14		usgrf1oedg.i	. . . . . . . . . . . 12 ⊢ 𝐼 = (iEdg‘𝐺)
15	14	eqcomi 2619	. . . . . . . . . . 11 ⊢ (iEdg‘𝐺) = 𝐼
16	15	a1i 11	. . . . . . . . . 10 ⊢ (𝐺 ∈ UHGraph → (iEdg‘𝐺) = 𝐼)
17	16	rneqd 5274	. . . . . . . . 9 ⊢ (𝐺 ∈ UHGraph → ran (iEdg‘𝐺) = ran 𝐼)
18	12, 13, 17	3eqtrd 2648	. . . . . . . 8 ⊢ (𝐺 ∈ UHGraph → 𝐸 = ran 𝐼)
19	18	eleq2d 2673	. . . . . . 7 ⊢ (𝐺 ∈ UHGraph → ({𝑁, 𝐴} ∈ 𝐸 ↔ {𝑁, 𝐴} ∈ ran 𝐼))
20	18	eleq2d 2673	. . . . . . 7 ⊢ (𝐺 ∈ UHGraph → ({𝐵, 𝑁} ∈ 𝐸 ↔ {𝐵, 𝑁} ∈ ran 𝐼))
21	19, 20	anbi12d 743	. . . . . 6 ⊢ (𝐺 ∈ UHGraph → (({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸) ↔ ({𝑁, 𝐴} ∈ ran 𝐼 ∧ {𝐵, 𝑁} ∈ ran 𝐼)))
22		eqid 2610	. . . . . . . . . 10 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
23	22	uhgrfun 25732	. . . . . . . . 9 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
24	14	funeqi 5824	. . . . . . . . 9 ⊢ (Fun 𝐼 ↔ Fun (iEdg‘𝐺))
25	23, 24	sylibr 223	. . . . . . . 8 ⊢ (𝐺 ∈ UHGraph → Fun 𝐼)
26		funfn 5833	. . . . . . . 8 ⊢ (Fun 𝐼 ↔ 𝐼 Fn dom 𝐼)
27	25, 26	sylib 207	. . . . . . 7 ⊢ (𝐺 ∈ UHGraph → 𝐼 Fn dom 𝐼)
28		fvelrnb 6153	. . . . . . . 8 ⊢ (𝐼 Fn dom 𝐼 → ({𝑁, 𝐴} ∈ ran 𝐼 ↔ ∃𝑥 ∈ dom 𝐼(𝐼‘𝑥) = {𝑁, 𝐴}))
29		fvelrnb 6153	. . . . . . . 8 ⊢ (𝐼 Fn dom 𝐼 → ({𝐵, 𝑁} ∈ ran 𝐼 ↔ ∃𝑦 ∈ dom 𝐼(𝐼‘𝑦) = {𝐵, 𝑁}))
30	28, 29	anbi12d 743	. . . . . . 7 ⊢ (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})))
31	27, 30	syl 17	. . . . . 6 ⊢ (𝐺 ∈ UHGraph → (({𝑁, 𝐴} ∈ ran 𝐼 ∧ {𝐵, 𝑁} ∈ ran 𝐼) ↔ (∃𝑥 ∈ dom 𝐼(𝐼‘𝑥) = {𝑁, 𝐴} ∧ ∃𝑦 ∈ dom 𝐼(𝐼‘𝑦) = {𝐵, 𝑁})))
32	21, 31	bitrd 267	. . . . 5 ⊢ (𝐺 ∈ UHGraph → (({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸) ↔ (∃𝑥 ∈ dom 𝐼(𝐼‘𝑥) = {𝑁, 𝐴} ∧ ∃𝑦 ∈ dom 𝐼(𝐼‘𝑦) = {𝐵, 𝑁})))
33	32	ad2antrr 758	. . . 4 ⊢ (((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})))
34		reeanv 3086	. . . . 5 ⊢ (∃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}))
35		fveq2 6103	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → (𝐼‘𝑥) = (𝐼‘𝑦))
36	35	eqeq1d 2612	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → ((𝐼‘𝑥) = {𝑁, 𝐴} ↔ (𝐼‘𝑦) = {𝑁, 𝐴}))
37	36	anbi1d 737	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (((𝐼‘𝑥) = {𝑁, 𝐴} ∧ (𝐼‘𝑦) = {𝐵, 𝑁}) ↔ ((𝐼‘𝑦) = {𝑁, 𝐴} ∧ (𝐼‘𝑦) = {𝐵, 𝑁})))
38		eqtr2 2630	. . . . . . . . . . . . . 14 ⊢ (((𝐼‘𝑦) = {𝑁, 𝐴} ∧ (𝐼‘𝑦) = {𝐵, 𝑁}) → {𝑁, 𝐴} = {𝐵, 𝑁})
39		prcom 4211	. . . . . . . . . . . . . . . 16 ⊢ {𝐵, 𝑁} = {𝑁, 𝐵}
40	39	eqeq2i 2622	. . . . . . . . . . . . . . 15 ⊢ ({𝑁, 𝐴} = {𝐵, 𝑁} ↔ {𝑁, 𝐴} = {𝑁, 𝐵})
41		preq12bg 4326	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ (𝑁 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → ({𝑁, 𝐴} = {𝑁, 𝐵} ↔ ((𝑁 = 𝑁 ∧ 𝐴 = 𝐵) ∨ (𝑁 = 𝐵 ∧ 𝐴 = 𝑁))))
42	41	ancom2s 840	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉)) → ({𝑁, 𝐴} = {𝑁, 𝐵} ↔ ((𝑁 = 𝑁 ∧ 𝐴 = 𝐵) ∨ (𝑁 = 𝐵 ∧ 𝐴 = 𝑁))))
43		eqneqall 2793	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → 𝑥 ≠ 𝑦))
44	43	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 = 𝑁 ∧ 𝐴 = 𝐵) → (𝐴 ≠ 𝐵 → 𝑥 ≠ 𝑦))
45		eqtr 2629	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 = 𝑁 ∧ 𝑁 = 𝐵) → 𝐴 = 𝐵)
46	45	ancoms 468	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑁 = 𝐵 ∧ 𝐴 = 𝑁) → 𝐴 = 𝐵)
47	46, 43	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 = 𝐵 ∧ 𝐴 = 𝑁) → (𝐴 ≠ 𝐵 → 𝑥 ≠ 𝑦))
48	44, 47	jaoi 393	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 = 𝑁 ∧ 𝐴 = 𝐵) ∨ (𝑁 = 𝐵 ∧ 𝐴 = 𝑁)) → (𝐴 ≠ 𝐵 → 𝑥 ≠ 𝑦))
49	48	adantld 482	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 = 𝑁 ∧ 𝐴 = 𝐵) ∨ (𝑁 = 𝐵 ∧ 𝐴 = 𝑁)) → ((𝐺 ∈ UHGraph ∧ 𝐴 ≠ 𝐵) → 𝑥 ≠ 𝑦))
50	42, 49	syl6bi 242	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉)) → ({𝑁, 𝐴} = {𝑁, 𝐵} → ((𝐺 ∈ UHGraph ∧ 𝐴 ≠ 𝐵) → 𝑥 ≠ 𝑦)))
51	50	com3l 87	. . . . . . . . . . . . . . . 16 ⊢ ({𝑁, 𝐴} = {𝑁, 𝐵} → ((𝐺 ∈ UHGraph ∧ 𝐴 ≠ 𝐵) → (((𝑁 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉)) → 𝑥 ≠ 𝑦)))
52	51	impd 446	. . . . . . . . . . . . . . 15 ⊢ ({𝑁, 𝐴} = {𝑁, 𝐵} → (((𝐺 ∈ UHGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑁 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))) → 𝑥 ≠ 𝑦))
53	40, 52	sylbi 206	. . . . . . . . . . . . . 14 ⊢ ({𝑁, 𝐴} = {𝐵, 𝑁} → (((𝐺 ∈ UHGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑁 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))) → 𝑥 ≠ 𝑦))
54	38, 53	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝐼‘𝑦) = {𝑁, 𝐴} ∧ (𝐼‘𝑦) = {𝐵, 𝑁}) → (((𝐺 ∈ UHGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑁 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))) → 𝑥 ≠ 𝑦))
55	37, 54	syl6bi 242	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (((𝐼‘𝑥) = {𝑁, 𝐴} ∧ (𝐼‘𝑦) = {𝐵, 𝑁}) → (((𝐺 ∈ UHGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑁 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))) → 𝑥 ≠ 𝑦)))
56	55	com23 84	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (((𝐺 ∈ UHGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑁 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))) → (((𝐼‘𝑥) = {𝑁, 𝐴} ∧ (𝐼‘𝑦) = {𝐵, 𝑁}) → 𝑥 ≠ 𝑦)))
57	56	impd 446	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ((((𝐺 ∈ UHGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑁 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))) ∧ ((𝐼‘𝑥) = {𝑁, 𝐴} ∧ (𝐼‘𝑦) = {𝐵, 𝑁})) → 𝑥 ≠ 𝑦))
58		ax-1 6	. . . . . . . . . 10 ⊢ (𝑥 ≠ 𝑦 → ((((𝐺 ∈ UHGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑁 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))) ∧ ((𝐼‘𝑥) = {𝑁, 𝐴} ∧ (𝐼‘𝑦) = {𝐵, 𝑁})) → 𝑥 ≠ 𝑦))
59	57, 58	pm2.61ine 2865	. . . . . . . . 9 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑁 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))) ∧ ((𝐼‘𝑥) = {𝑁, 𝐴} ∧ (𝐼‘𝑦) = {𝐵, 𝑁})) → 𝑥 ≠ 𝑦)
60		prid1g 4239	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ 𝑉 → 𝑁 ∈ {𝑁, 𝐴})
61	60	ad2antrr 758	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉)) → 𝑁 ∈ {𝑁, 𝐴})
62	61	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑁 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))) → 𝑁 ∈ {𝑁, 𝐴})
63		eleq2 2677	. . . . . . . . . . . 12 ⊢ ((𝐼‘𝑥) = {𝑁, 𝐴} → (𝑁 ∈ (𝐼‘𝑥) ↔ 𝑁 ∈ {𝑁, 𝐴}))
64	62, 63	syl5ibr 235	. . . . . . . . . . 11 ⊢ ((𝐼‘𝑥) = {𝑁, 𝐴} → (((𝐺 ∈ UHGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑁 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))) → 𝑁 ∈ (𝐼‘𝑥)))
65	64	adantr 480	. . . . . . . . . 10 ⊢ (((𝐼‘𝑥) = {𝑁, 𝐴} ∧ (𝐼‘𝑦) = {𝐵, 𝑁}) → (((𝐺 ∈ UHGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑁 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))) → 𝑁 ∈ (𝐼‘𝑥)))
66	65	impcom 445	. . . . . . . . 9 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑁 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))) ∧ ((𝐼‘𝑥) = {𝑁, 𝐴} ∧ (𝐼‘𝑦) = {𝐵, 𝑁})) → 𝑁 ∈ (𝐼‘𝑥))
67		prid2g 4240	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ 𝑉 → 𝑁 ∈ {𝐵, 𝑁})
68	67	ad2antrr 758	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉)) → 𝑁 ∈ {𝐵, 𝑁})
69	68	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑁 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))) → 𝑁 ∈ {𝐵, 𝑁})
70		eleq2 2677	. . . . . . . . . . . 12 ⊢ ((𝐼‘𝑦) = {𝐵, 𝑁} → (𝑁 ∈ (𝐼‘𝑦) ↔ 𝑁 ∈ {𝐵, 𝑁}))
71	69, 70	syl5ibr 235	. . . . . . . . . . 11 ⊢ ((𝐼‘𝑦) = {𝐵, 𝑁} → (((𝐺 ∈ UHGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑁 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))) → 𝑁 ∈ (𝐼‘𝑦)))
72	71	adantl 481	. . . . . . . . . 10 ⊢ (((𝐼‘𝑥) = {𝑁, 𝐴} ∧ (𝐼‘𝑦) = {𝐵, 𝑁}) → (((𝐺 ∈ UHGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑁 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))) → 𝑁 ∈ (𝐼‘𝑦)))
73	72	impcom 445	. . . . . . . . 9 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑁 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))) ∧ ((𝐼‘𝑥) = {𝑁, 𝐴} ∧ (𝐼‘𝑦) = {𝐵, 𝑁})) → 𝑁 ∈ (𝐼‘𝑦))
74	59, 66, 73	3jca 1235	. . . . . . . 8 ⊢ ((((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)))
75	74	ex 449	. . . . . . 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))))
76	75	reximdv 2999	. . . . . 6 ⊢ (((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))))
77	76	reximdv 2999	. . . . 5 ⊢ (((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))))
78	34, 77	syl5bir 232	. . . 4 ⊢ (((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))))
79	33, 78	sylbid 229	. . 3 ⊢ (((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))))
80	79	imp 444	. 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)))
81	10, 80	syl 17	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   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897  {cpr 4127  dom cdm 5038  ran crn 5039  Fun wfun 5798   Fn wfn 5799  ‘cfv 5804  Vtxcvtx 25673  iEdgciedg 25674   UHGraph cuhgr 25722  Edgcedga 25792

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-uhgr 25724  df-edga 25793

This theorem is referenced by:  umgr2edg  40436