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Theorem frgrnbnb 30241

Description: If two neighbors 𝑈 and 𝑊 of a vertex 𝑋 have a common neighbor 𝐴 in a friendship graph, then this common neighbor 𝐴 must be the vertex 𝑋. (Contributed by Alexander van der Vekens, 19-Dec-2017.) (Revised by AV, 2-Apr-2021.) (Proof shortened by AV, 13-Feb-2022.)

**Hypotheses**
Ref	Expression
frgrnbnb.e	⊢ 𝐸 = (Edg‘𝐺)
frgrnbnb.n	⊢ 𝐷 = (𝐺 NeighbVtx 𝑋)

**Assertion**
Ref	Expression
frgrnbnb	⊢ ((𝐺 ∈ FriendGraph ∧ (𝑈 ∈ 𝐷 ∧ 𝑊 ∈ 𝐷) ∧ 𝑈 ≠ 𝑊) → (({𝑈, 𝐴} ∈ 𝐸 ∧ {𝑊, 𝐴} ∈ 𝐸) → 𝐴 = 𝑋))

**Proof of Theorem frgrnbnb**
Step	Hyp	Ref	Expression
1		frgrusgr 30209	. . 3 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
2		frgrnbnb.n	. . . . . . . . . 10 ⊢ 𝐷 = (𝐺 NeighbVtx 𝑋)
3	2	eleq2i 2820	. . . . . . . . 9 ⊢ (𝑈 ∈ 𝐷 ↔ 𝑈 ∈ (𝐺 NeighbVtx 𝑋))
4		frgrnbnb.e	. . . . . . . . . . 11 ⊢ 𝐸 = (Edg‘𝐺)
5	4	nbusgreledg 29302	. . . . . . . . . 10 ⊢ (𝐺 ∈ USGraph → (𝑈 ∈ (𝐺 NeighbVtx 𝑋) ↔ {𝑈, 𝑋} ∈ 𝐸))
6	5	biimpd 229	. . . . . . . . 9 ⊢ (𝐺 ∈ USGraph → (𝑈 ∈ (𝐺 NeighbVtx 𝑋) → {𝑈, 𝑋} ∈ 𝐸))
7	3, 6	biimtrid 242	. . . . . . . 8 ⊢ (𝐺 ∈ USGraph → (𝑈 ∈ 𝐷 → {𝑈, 𝑋} ∈ 𝐸))
8	2	eleq2i 2820	. . . . . . . . 9 ⊢ (𝑊 ∈ 𝐷 ↔ 𝑊 ∈ (𝐺 NeighbVtx 𝑋))
9	4	nbusgreledg 29302	. . . . . . . . . 10 ⊢ (𝐺 ∈ USGraph → (𝑊 ∈ (𝐺 NeighbVtx 𝑋) ↔ {𝑊, 𝑋} ∈ 𝐸))
10	9	biimpd 229	. . . . . . . . 9 ⊢ (𝐺 ∈ USGraph → (𝑊 ∈ (𝐺 NeighbVtx 𝑋) → {𝑊, 𝑋} ∈ 𝐸))
11	8, 10	biimtrid 242	. . . . . . . 8 ⊢ (𝐺 ∈ USGraph → (𝑊 ∈ 𝐷 → {𝑊, 𝑋} ∈ 𝐸))
12	7, 11	anim12d 609	. . . . . . 7 ⊢ (𝐺 ∈ USGraph → ((𝑈 ∈ 𝐷 ∧ 𝑊 ∈ 𝐷) → ({𝑈, 𝑋} ∈ 𝐸 ∧ {𝑊, 𝑋} ∈ 𝐸)))
13	12	imp 406	. . . . . 6 ⊢ ((𝐺 ∈ USGraph ∧ (𝑈 ∈ 𝐷 ∧ 𝑊 ∈ 𝐷)) → ({𝑈, 𝑋} ∈ 𝐸 ∧ {𝑊, 𝑋} ∈ 𝐸))
14		eqid 2729	. . . . . . . . . 10 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
15	14	nbgrisvtx 29290	. . . . . . . . 9 ⊢ (𝑈 ∈ (𝐺 NeighbVtx 𝑋) → 𝑈 ∈ (Vtx‘𝐺))
16	15, 2	eleq2s 2846	. . . . . . . 8 ⊢ (𝑈 ∈ 𝐷 → 𝑈 ∈ (Vtx‘𝐺))
17	14	nbgrisvtx 29290	. . . . . . . . 9 ⊢ (𝑊 ∈ (𝐺 NeighbVtx 𝑋) → 𝑊 ∈ (Vtx‘𝐺))
18	17, 2	eleq2s 2846	. . . . . . . 8 ⊢ (𝑊 ∈ 𝐷 → 𝑊 ∈ (Vtx‘𝐺))
19	16, 18	anim12i 613	. . . . . . 7 ⊢ ((𝑈 ∈ 𝐷 ∧ 𝑊 ∈ 𝐷) → (𝑈 ∈ (Vtx‘𝐺) ∧ 𝑊 ∈ (Vtx‘𝐺)))
20	19	adantl 481	. . . . . 6 ⊢ ((𝐺 ∈ USGraph ∧ (𝑈 ∈ 𝐷 ∧ 𝑊 ∈ 𝐷)) → (𝑈 ∈ (Vtx‘𝐺) ∧ 𝑊 ∈ (Vtx‘𝐺)))
21	4, 14	usgrpredgv 29146	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ USGraph ∧ {𝑈, 𝐴} ∈ 𝐸) → (𝑈 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)))
22	21	ad2ant2r 747	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ USGraph ∧ (𝑈 ∈ 𝐷 ∧ 𝑊 ∈ 𝐷)) ∧ ({𝑈, 𝐴} ∈ 𝐸 ∧ {𝑊, 𝐴} ∈ 𝐸)) → (𝑈 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)))
23		ax-1 6	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐴 = 𝑋 → (𝐺 ∈ FriendGraph → 𝐴 = 𝑋))
24	23	2a1d 26	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐴 = 𝑋 → (({𝑈, 𝐴} ∈ 𝐸 ∧ {𝑊, 𝐴} ∈ 𝐸) → (((𝑋 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)) ∧ (𝑈 ∈ (Vtx‘𝐺) ∧ 𝑊 ∈ (Vtx‘𝐺))) → (𝐺 ∈ FriendGraph → 𝐴 = 𝑋))))
25	24	2a1d 26	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐴 = 𝑋 → (𝑈 ≠ 𝑊 → (({𝑈, 𝑋} ∈ 𝐸 ∧ {𝑊, 𝑋} ∈ 𝐸) → (({𝑈, 𝐴} ∈ 𝐸 ∧ {𝑊, 𝐴} ∈ 𝐸) → (((𝑋 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)) ∧ (𝑈 ∈ (Vtx‘𝐺) ∧ 𝑊 ∈ (Vtx‘𝐺))) → (𝐺 ∈ FriendGraph → 𝐴 = 𝑋))))))
26		simpll 766	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝐺 ∈ USGraph ∧ ((𝑋 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)) ∧ (𝑈 ∈ (Vtx‘𝐺) ∧ 𝑊 ∈ (Vtx‘𝐺)))) ∧ (𝐴 ≠ 𝑋 ∧ 𝑈 ≠ 𝑊)) → 𝐺 ∈ USGraph)
27		simprrr 781	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝐺 ∈ USGraph ∧ ((𝑋 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)) ∧ (𝑈 ∈ (Vtx‘𝐺) ∧ 𝑊 ∈ (Vtx‘𝐺)))) → 𝑊 ∈ (Vtx‘𝐺))
28	27	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝐺 ∈ USGraph ∧ ((𝑋 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)) ∧ (𝑈 ∈ (Vtx‘𝐺) ∧ 𝑊 ∈ (Vtx‘𝐺)))) ∧ (𝐴 ≠ 𝑋 ∧ 𝑈 ≠ 𝑊)) → 𝑊 ∈ (Vtx‘𝐺))
29		simprrl 780	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝐺 ∈ USGraph ∧ ((𝑋 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)) ∧ (𝑈 ∈ (Vtx‘𝐺) ∧ 𝑊 ∈ (Vtx‘𝐺)))) → 𝑈 ∈ (Vtx‘𝐺))
30	29	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝐺 ∈ USGraph ∧ ((𝑋 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)) ∧ (𝑈 ∈ (Vtx‘𝐺) ∧ 𝑊 ∈ (Vtx‘𝐺)))) ∧ (𝐴 ≠ 𝑋 ∧ 𝑈 ≠ 𝑊)) → 𝑈 ∈ (Vtx‘𝐺))
31		necom 2978	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑈 ≠ 𝑊 ↔ 𝑊 ≠ 𝑈)
32	31	biimpi 216	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑈 ≠ 𝑊 → 𝑊 ≠ 𝑈)
33	32	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝐴 ≠ 𝑋 ∧ 𝑈 ≠ 𝑊) → 𝑊 ≠ 𝑈)
34	33	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝐺 ∈ USGraph ∧ ((𝑋 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)) ∧ (𝑈 ∈ (Vtx‘𝐺) ∧ 𝑊 ∈ (Vtx‘𝐺)))) ∧ (𝐴 ≠ 𝑋 ∧ 𝑈 ≠ 𝑊)) → 𝑊 ≠ 𝑈)
35	28, 30, 34	3jca 1128	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝐺 ∈ USGraph ∧ ((𝑋 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)) ∧ (𝑈 ∈ (Vtx‘𝐺) ∧ 𝑊 ∈ (Vtx‘𝐺)))) ∧ (𝐴 ≠ 𝑋 ∧ 𝑈 ≠ 𝑊)) → (𝑊 ∈ (Vtx‘𝐺) ∧ 𝑈 ∈ (Vtx‘𝐺) ∧ 𝑊 ≠ 𝑈))
36		simprll 778	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝐺 ∈ USGraph ∧ ((𝑋 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)) ∧ (𝑈 ∈ (Vtx‘𝐺) ∧ 𝑊 ∈ (Vtx‘𝐺)))) → 𝑋 ∈ (Vtx‘𝐺))
37	36	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝐺 ∈ USGraph ∧ ((𝑋 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)) ∧ (𝑈 ∈ (Vtx‘𝐺) ∧ 𝑊 ∈ (Vtx‘𝐺)))) ∧ (𝐴 ≠ 𝑋 ∧ 𝑈 ≠ 𝑊)) → 𝑋 ∈ (Vtx‘𝐺))
38		simprlr 779	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝐺 ∈ USGraph ∧ ((𝑋 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)) ∧ (𝑈 ∈ (Vtx‘𝐺) ∧ 𝑊 ∈ (Vtx‘𝐺)))) → 𝐴 ∈ (Vtx‘𝐺))
39	38	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝐺 ∈ USGraph ∧ ((𝑋 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)) ∧ (𝑈 ∈ (Vtx‘𝐺) ∧ 𝑊 ∈ (Vtx‘𝐺)))) ∧ (𝐴 ≠ 𝑋 ∧ 𝑈 ≠ 𝑊)) → 𝐴 ∈ (Vtx‘𝐺))
40		necom 2978	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝐴 ≠ 𝑋 ↔ 𝑋 ≠ 𝐴)
41	40	biimpi 216	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝐴 ≠ 𝑋 → 𝑋 ≠ 𝐴)
42	41	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝐴 ≠ 𝑋 ∧ 𝑈 ≠ 𝑊) → 𝑋 ≠ 𝐴)
43	42	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝐺 ∈ USGraph ∧ ((𝑋 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)) ∧ (𝑈 ∈ (Vtx‘𝐺) ∧ 𝑊 ∈ (Vtx‘𝐺)))) ∧ (𝐴 ≠ 𝑋 ∧ 𝑈 ≠ 𝑊)) → 𝑋 ≠ 𝐴)
44	37, 39, 43	3jca 1128	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝐺 ∈ USGraph ∧ ((𝑋 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)) ∧ (𝑈 ∈ (Vtx‘𝐺) ∧ 𝑊 ∈ (Vtx‘𝐺)))) ∧ (𝐴 ≠ 𝑋 ∧ 𝑈 ≠ 𝑊)) → (𝑋 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺) ∧ 𝑋 ≠ 𝐴))
45	26, 35, 44	3jca 1128	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝐺 ∈ USGraph ∧ ((𝑋 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)) ∧ (𝑈 ∈ (Vtx‘𝐺) ∧ 𝑊 ∈ (Vtx‘𝐺)))) ∧ (𝐴 ≠ 𝑋 ∧ 𝑈 ≠ 𝑊)) → (𝐺 ∈ USGraph ∧ (𝑊 ∈ (Vtx‘𝐺) ∧ 𝑈 ∈ (Vtx‘𝐺) ∧ 𝑊 ≠ 𝑈) ∧ (𝑋 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺) ∧ 𝑋 ≠ 𝐴)))
46	45	ad4ant14 752	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((((𝐺 ∈ USGraph ∧ ((𝑋 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)) ∧ (𝑈 ∈ (Vtx‘𝐺) ∧ 𝑊 ∈ (Vtx‘𝐺)))) ∧ ({𝑈, 𝑋} ∈ 𝐸 ∧ {𝑊, 𝑋} ∈ 𝐸)) ∧ ({𝑈, 𝐴} ∈ 𝐸 ∧ {𝑊, 𝐴} ∈ 𝐸)) ∧ (𝐴 ≠ 𝑋 ∧ 𝑈 ≠ 𝑊)) → (𝐺 ∈ USGraph ∧ (𝑊 ∈ (Vtx‘𝐺) ∧ 𝑈 ∈ (Vtx‘𝐺) ∧ 𝑊 ≠ 𝑈) ∧ (𝑋 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺) ∧ 𝑋 ≠ 𝐴)))
47		prcom 4684	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ {𝑈, 𝑋} = {𝑋, 𝑈}
48	47	eleq1i 2819	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ({𝑈, 𝑋} ∈ 𝐸 ↔ {𝑋, 𝑈} ∈ 𝐸)
49	48	biimpi 216	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ({𝑈, 𝑋} ∈ 𝐸 → {𝑋, 𝑈} ∈ 𝐸)
50	49	anim1ci 616	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (({𝑈, 𝑋} ∈ 𝐸 ∧ {𝑊, 𝑋} ∈ 𝐸) → ({𝑊, 𝑋} ∈ 𝐸 ∧ {𝑋, 𝑈} ∈ 𝐸))
51	50	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝐺 ∈ USGraph ∧ ((𝑋 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)) ∧ (𝑈 ∈ (Vtx‘𝐺) ∧ 𝑊 ∈ (Vtx‘𝐺)))) ∧ ({𝑈, 𝑋} ∈ 𝐸 ∧ {𝑊, 𝑋} ∈ 𝐸)) → ({𝑊, 𝑋} ∈ 𝐸 ∧ {𝑋, 𝑈} ∈ 𝐸))
52		prcom 4684	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ {𝑊, 𝐴} = {𝐴, 𝑊}
53	52	eleq1i 2819	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ({𝑊, 𝐴} ∈ 𝐸 ↔ {𝐴, 𝑊} ∈ 𝐸)
54	53	biimpi 216	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ({𝑊, 𝐴} ∈ 𝐸 → {𝐴, 𝑊} ∈ 𝐸)
55	54	anim2i 617	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (({𝑈, 𝐴} ∈ 𝐸 ∧ {𝑊, 𝐴} ∈ 𝐸) → ({𝑈, 𝐴} ∈ 𝐸 ∧ {𝐴, 𝑊} ∈ 𝐸))
56	51, 55	anim12i 613	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝐺 ∈ USGraph ∧ ((𝑋 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)) ∧ (𝑈 ∈ (Vtx‘𝐺) ∧ 𝑊 ∈ (Vtx‘𝐺)))) ∧ ({𝑈, 𝑋} ∈ 𝐸 ∧ {𝑊, 𝑋} ∈ 𝐸)) ∧ ({𝑈, 𝐴} ∈ 𝐸 ∧ {𝑊, 𝐴} ∈ 𝐸)) → (({𝑊, 𝑋} ∈ 𝐸 ∧ {𝑋, 𝑈} ∈ 𝐸) ∧ ({𝑈, 𝐴} ∈ 𝐸 ∧ {𝐴, 𝑊} ∈ 𝐸)))
57	56	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((((𝐺 ∈ USGraph ∧ ((𝑋 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)) ∧ (𝑈 ∈ (Vtx‘𝐺) ∧ 𝑊 ∈ (Vtx‘𝐺)))) ∧ ({𝑈, 𝑋} ∈ 𝐸 ∧ {𝑊, 𝑋} ∈ 𝐸)) ∧ ({𝑈, 𝐴} ∈ 𝐸 ∧ {𝑊, 𝐴} ∈ 𝐸)) ∧ (𝐴 ≠ 𝑋 ∧ 𝑈 ≠ 𝑊)) → (({𝑊, 𝑋} ∈ 𝐸 ∧ {𝑋, 𝑈} ∈ 𝐸) ∧ ({𝑈, 𝐴} ∈ 𝐸 ∧ {𝐴, 𝑊} ∈ 𝐸)))
58	14, 4	4cyclusnfrgr 30240	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝐺 ∈ USGraph ∧ (𝑊 ∈ (Vtx‘𝐺) ∧ 𝑈 ∈ (Vtx‘𝐺) ∧ 𝑊 ≠ 𝑈) ∧ (𝑋 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺) ∧ 𝑋 ≠ 𝐴)) → ((({𝑊, 𝑋} ∈ 𝐸 ∧ {𝑋, 𝑈} ∈ 𝐸) ∧ ({𝑈, 𝐴} ∈ 𝐸 ∧ {𝐴, 𝑊} ∈ 𝐸)) → 𝐺 ∉ FriendGraph ))
59	46, 57, 58	sylc 65	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((((𝐺 ∈ USGraph ∧ ((𝑋 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)) ∧ (𝑈 ∈ (Vtx‘𝐺) ∧ 𝑊 ∈ (Vtx‘𝐺)))) ∧ ({𝑈, 𝑋} ∈ 𝐸 ∧ {𝑊, 𝑋} ∈ 𝐸)) ∧ ({𝑈, 𝐴} ∈ 𝐸 ∧ {𝑊, 𝐴} ∈ 𝐸)) ∧ (𝐴 ≠ 𝑋 ∧ 𝑈 ≠ 𝑊)) → 𝐺 ∉ FriendGraph )
60		df-nel 3030	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝐺 ∉ FriendGraph ↔ ¬ 𝐺 ∈ FriendGraph )
61	59, 60	sylib 218	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((((𝐺 ∈ USGraph ∧ ((𝑋 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)) ∧ (𝑈 ∈ (Vtx‘𝐺) ∧ 𝑊 ∈ (Vtx‘𝐺)))) ∧ ({𝑈, 𝑋} ∈ 𝐸 ∧ {𝑊, 𝑋} ∈ 𝐸)) ∧ ({𝑈, 𝐴} ∈ 𝐸 ∧ {𝑊, 𝐴} ∈ 𝐸)) ∧ (𝐴 ≠ 𝑋 ∧ 𝑈 ≠ 𝑊)) → ¬ 𝐺 ∈ FriendGraph )
62	61	pm2.21d 121	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((((𝐺 ∈ USGraph ∧ ((𝑋 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)) ∧ (𝑈 ∈ (Vtx‘𝐺) ∧ 𝑊 ∈ (Vtx‘𝐺)))) ∧ ({𝑈, 𝑋} ∈ 𝐸 ∧ {𝑊, 𝑋} ∈ 𝐸)) ∧ ({𝑈, 𝐴} ∈ 𝐸 ∧ {𝑊, 𝐴} ∈ 𝐸)) ∧ (𝐴 ≠ 𝑋 ∧ 𝑈 ≠ 𝑊)) → (𝐺 ∈ FriendGraph → 𝐴 = 𝑋))
63	62	ex 412	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝐺 ∈ USGraph ∧ ((𝑋 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)) ∧ (𝑈 ∈ (Vtx‘𝐺) ∧ 𝑊 ∈ (Vtx‘𝐺)))) ∧ ({𝑈, 𝑋} ∈ 𝐸 ∧ {𝑊, 𝑋} ∈ 𝐸)) ∧ ({𝑈, 𝐴} ∈ 𝐸 ∧ {𝑊, 𝐴} ∈ 𝐸)) → ((𝐴 ≠ 𝑋 ∧ 𝑈 ≠ 𝑊) → (𝐺 ∈ FriendGraph → 𝐴 = 𝑋)))
64	63	com23 86	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝐺 ∈ USGraph ∧ ((𝑋 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)) ∧ (𝑈 ∈ (Vtx‘𝐺) ∧ 𝑊 ∈ (Vtx‘𝐺)))) ∧ ({𝑈, 𝑋} ∈ 𝐸 ∧ {𝑊, 𝑋} ∈ 𝐸)) ∧ ({𝑈, 𝐴} ∈ 𝐸 ∧ {𝑊, 𝐴} ∈ 𝐸)) → (𝐺 ∈ FriendGraph → ((𝐴 ≠ 𝑋 ∧ 𝑈 ≠ 𝑊) → 𝐴 = 𝑋)))
65	64	exp41 434	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐺 ∈ USGraph → (((𝑋 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)) ∧ (𝑈 ∈ (Vtx‘𝐺) ∧ 𝑊 ∈ (Vtx‘𝐺))) → (({𝑈, 𝑋} ∈ 𝐸 ∧ {𝑊, 𝑋} ∈ 𝐸) → (({𝑈, 𝐴} ∈ 𝐸 ∧ {𝑊, 𝐴} ∈ 𝐸) → (𝐺 ∈ FriendGraph → ((𝐴 ≠ 𝑋 ∧ 𝑈 ≠ 𝑊) → 𝐴 = 𝑋))))))
66	65	com25 99	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐺 ∈ USGraph → (𝐺 ∈ FriendGraph → (({𝑈, 𝑋} ∈ 𝐸 ∧ {𝑊, 𝑋} ∈ 𝐸) → (({𝑈, 𝐴} ∈ 𝐸 ∧ {𝑊, 𝐴} ∈ 𝐸) → (((𝑋 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)) ∧ (𝑈 ∈ (Vtx‘𝐺) ∧ 𝑊 ∈ (Vtx‘𝐺))) → ((𝐴 ≠ 𝑋 ∧ 𝑈 ≠ 𝑊) → 𝐴 = 𝑋))))))
67	1, 66	mpcom 38	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐺 ∈ FriendGraph → (({𝑈, 𝑋} ∈ 𝐸 ∧ {𝑊, 𝑋} ∈ 𝐸) → (({𝑈, 𝐴} ∈ 𝐸 ∧ {𝑊, 𝐴} ∈ 𝐸) → (((𝑋 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)) ∧ (𝑈 ∈ (Vtx‘𝐺) ∧ 𝑊 ∈ (Vtx‘𝐺))) → ((𝐴 ≠ 𝑋 ∧ 𝑈 ≠ 𝑊) → 𝐴 = 𝑋)))))
68	67	com15 101	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ≠ 𝑋 ∧ 𝑈 ≠ 𝑊) → (({𝑈, 𝑋} ∈ 𝐸 ∧ {𝑊, 𝑋} ∈ 𝐸) → (({𝑈, 𝐴} ∈ 𝐸 ∧ {𝑊, 𝐴} ∈ 𝐸) → (((𝑋 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)) ∧ (𝑈 ∈ (Vtx‘𝐺) ∧ 𝑊 ∈ (Vtx‘𝐺))) → (𝐺 ∈ FriendGraph → 𝐴 = 𝑋)))))
69	68	ex 412	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐴 ≠ 𝑋 → (𝑈 ≠ 𝑊 → (({𝑈, 𝑋} ∈ 𝐸 ∧ {𝑊, 𝑋} ∈ 𝐸) → (({𝑈, 𝐴} ∈ 𝐸 ∧ {𝑊, 𝐴} ∈ 𝐸) → (((𝑋 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)) ∧ (𝑈 ∈ (Vtx‘𝐺) ∧ 𝑊 ∈ (Vtx‘𝐺))) → (𝐺 ∈ FriendGraph → 𝐴 = 𝑋))))))
70	25, 69	pm2.61ine 3008	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑈 ≠ 𝑊 → (({𝑈, 𝑋} ∈ 𝐸 ∧ {𝑊, 𝑋} ∈ 𝐸) → (({𝑈, 𝐴} ∈ 𝐸 ∧ {𝑊, 𝐴} ∈ 𝐸) → (((𝑋 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)) ∧ (𝑈 ∈ (Vtx‘𝐺) ∧ 𝑊 ∈ (Vtx‘𝐺))) → (𝐺 ∈ FriendGraph → 𝐴 = 𝑋)))))
71	70	imp 406	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑈 ≠ 𝑊 ∧ ({𝑈, 𝑋} ∈ 𝐸 ∧ {𝑊, 𝑋} ∈ 𝐸)) → (({𝑈, 𝐴} ∈ 𝐸 ∧ {𝑊, 𝐴} ∈ 𝐸) → (((𝑋 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)) ∧ (𝑈 ∈ (Vtx‘𝐺) ∧ 𝑊 ∈ (Vtx‘𝐺))) → (𝐺 ∈ FriendGraph → 𝐴 = 𝑋))))
72	71	com13 88	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑋 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)) ∧ (𝑈 ∈ (Vtx‘𝐺) ∧ 𝑊 ∈ (Vtx‘𝐺))) → (({𝑈, 𝐴} ∈ 𝐸 ∧ {𝑊, 𝐴} ∈ 𝐸) → ((𝑈 ≠ 𝑊 ∧ ({𝑈, 𝑋} ∈ 𝐸 ∧ {𝑊, 𝑋} ∈ 𝐸)) → (𝐺 ∈ FriendGraph → 𝐴 = 𝑋))))
73	72	ex 412	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑋 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)) → ((𝑈 ∈ (Vtx‘𝐺) ∧ 𝑊 ∈ (Vtx‘𝐺)) → (({𝑈, 𝐴} ∈ 𝐸 ∧ {𝑊, 𝐴} ∈ 𝐸) → ((𝑈 ≠ 𝑊 ∧ ({𝑈, 𝑋} ∈ 𝐸 ∧ {𝑊, 𝑋} ∈ 𝐸)) → (𝐺 ∈ FriendGraph → 𝐴 = 𝑋)))))
74	73	com25 99	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑋 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)) → (𝐺 ∈ FriendGraph → (({𝑈, 𝐴} ∈ 𝐸 ∧ {𝑊, 𝐴} ∈ 𝐸) → ((𝑈 ≠ 𝑊 ∧ ({𝑈, 𝑋} ∈ 𝐸 ∧ {𝑊, 𝑋} ∈ 𝐸)) → ((𝑈 ∈ (Vtx‘𝐺) ∧ 𝑊 ∈ (Vtx‘𝐺)) → 𝐴 = 𝑋)))))
75	74	ex 412	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 ∈ (Vtx‘𝐺) → (𝐴 ∈ (Vtx‘𝐺) → (𝐺 ∈ FriendGraph → (({𝑈, 𝐴} ∈ 𝐸 ∧ {𝑊, 𝐴} ∈ 𝐸) → ((𝑈 ≠ 𝑊 ∧ ({𝑈, 𝑋} ∈ 𝐸 ∧ {𝑊, 𝑋} ∈ 𝐸)) → ((𝑈 ∈ (Vtx‘𝐺) ∧ 𝑊 ∈ (Vtx‘𝐺)) → 𝐴 = 𝑋))))))
76	14	nbgrcl 29284	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑈 ∈ (𝐺 NeighbVtx 𝑋) → 𝑋 ∈ (Vtx‘𝐺))
77	76, 2	eleq2s 2846	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑈 ∈ 𝐷 → 𝑋 ∈ (Vtx‘𝐺))
78	77	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑈 ∈ 𝐷 ∧ 𝑊 ∈ 𝐷) → 𝑋 ∈ (Vtx‘𝐺))
79	78	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ∈ USGraph ∧ (𝑈 ∈ 𝐷 ∧ 𝑊 ∈ 𝐷)) → 𝑋 ∈ (Vtx‘𝐺))
80	75, 79	syl11 33	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ (Vtx‘𝐺) → ((𝐺 ∈ USGraph ∧ (𝑈 ∈ 𝐷 ∧ 𝑊 ∈ 𝐷)) → (𝐺 ∈ FriendGraph → (({𝑈, 𝐴} ∈ 𝐸 ∧ {𝑊, 𝐴} ∈ 𝐸) → ((𝑈 ≠ 𝑊 ∧ ({𝑈, 𝑋} ∈ 𝐸 ∧ {𝑊, 𝑋} ∈ 𝐸)) → ((𝑈 ∈ (Vtx‘𝐺) ∧ 𝑊 ∈ (Vtx‘𝐺)) → 𝐴 = 𝑋))))))
81	80	com34 91	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ (Vtx‘𝐺) → ((𝐺 ∈ USGraph ∧ (𝑈 ∈ 𝐷 ∧ 𝑊 ∈ 𝐷)) → (({𝑈, 𝐴} ∈ 𝐸 ∧ {𝑊, 𝐴} ∈ 𝐸) → (𝐺 ∈ FriendGraph → ((𝑈 ≠ 𝑊 ∧ ({𝑈, 𝑋} ∈ 𝐸 ∧ {𝑊, 𝑋} ∈ 𝐸)) → ((𝑈 ∈ (Vtx‘𝐺) ∧ 𝑊 ∈ (Vtx‘𝐺)) → 𝐴 = 𝑋))))))
82	81	impd 410	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ (Vtx‘𝐺) → (((𝐺 ∈ USGraph ∧ (𝑈 ∈ 𝐷 ∧ 𝑊 ∈ 𝐷)) ∧ ({𝑈, 𝐴} ∈ 𝐸 ∧ {𝑊, 𝐴} ∈ 𝐸)) → (𝐺 ∈ FriendGraph → ((𝑈 ≠ 𝑊 ∧ ({𝑈, 𝑋} ∈ 𝐸 ∧ {𝑊, 𝑋} ∈ 𝐸)) → ((𝑈 ∈ (Vtx‘𝐺) ∧ 𝑊 ∈ (Vtx‘𝐺)) → 𝐴 = 𝑋)))))
83	82	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)) → (((𝐺 ∈ USGraph ∧ (𝑈 ∈ 𝐷 ∧ 𝑊 ∈ 𝐷)) ∧ ({𝑈, 𝐴} ∈ 𝐸 ∧ {𝑊, 𝐴} ∈ 𝐸)) → (𝐺 ∈ FriendGraph → ((𝑈 ≠ 𝑊 ∧ ({𝑈, 𝑋} ∈ 𝐸 ∧ {𝑊, 𝑋} ∈ 𝐸)) → ((𝑈 ∈ (Vtx‘𝐺) ∧ 𝑊 ∈ (Vtx‘𝐺)) → 𝐴 = 𝑋)))))
84	22, 83	mpcom 38	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ USGraph ∧ (𝑈 ∈ 𝐷 ∧ 𝑊 ∈ 𝐷)) ∧ ({𝑈, 𝐴} ∈ 𝐸 ∧ {𝑊, 𝐴} ∈ 𝐸)) → (𝐺 ∈ FriendGraph → ((𝑈 ≠ 𝑊 ∧ ({𝑈, 𝑋} ∈ 𝐸 ∧ {𝑊, 𝑋} ∈ 𝐸)) → ((𝑈 ∈ (Vtx‘𝐺) ∧ 𝑊 ∈ (Vtx‘𝐺)) → 𝐴 = 𝑋))))
85	84	ex 412	. . . . . . . . . 10 ⊢ ((𝐺 ∈ USGraph ∧ (𝑈 ∈ 𝐷 ∧ 𝑊 ∈ 𝐷)) → (({𝑈, 𝐴} ∈ 𝐸 ∧ {𝑊, 𝐴} ∈ 𝐸) → (𝐺 ∈ FriendGraph → ((𝑈 ≠ 𝑊 ∧ ({𝑈, 𝑋} ∈ 𝐸 ∧ {𝑊, 𝑋} ∈ 𝐸)) → ((𝑈 ∈ (Vtx‘𝐺) ∧ 𝑊 ∈ (Vtx‘𝐺)) → 𝐴 = 𝑋)))))
86	85	com25 99	. . . . . . . . 9 ⊢ ((𝐺 ∈ USGraph ∧ (𝑈 ∈ 𝐷 ∧ 𝑊 ∈ 𝐷)) → ((𝑈 ∈ (Vtx‘𝐺) ∧ 𝑊 ∈ (Vtx‘𝐺)) → (𝐺 ∈ FriendGraph → ((𝑈 ≠ 𝑊 ∧ ({𝑈, 𝑋} ∈ 𝐸 ∧ {𝑊, 𝑋} ∈ 𝐸)) → (({𝑈, 𝐴} ∈ 𝐸 ∧ {𝑊, 𝐴} ∈ 𝐸) → 𝐴 = 𝑋)))))
87	86	com14 96	. . . . . . . 8 ⊢ ((𝑈 ≠ 𝑊 ∧ ({𝑈, 𝑋} ∈ 𝐸 ∧ {𝑊, 𝑋} ∈ 𝐸)) → ((𝑈 ∈ (Vtx‘𝐺) ∧ 𝑊 ∈ (Vtx‘𝐺)) → (𝐺 ∈ FriendGraph → ((𝐺 ∈ USGraph ∧ (𝑈 ∈ 𝐷 ∧ 𝑊 ∈ 𝐷)) → (({𝑈, 𝐴} ∈ 𝐸 ∧ {𝑊, 𝐴} ∈ 𝐸) → 𝐴 = 𝑋)))))
88	87	ex 412	. . . . . . 7 ⊢ (𝑈 ≠ 𝑊 → (({𝑈, 𝑋} ∈ 𝐸 ∧ {𝑊, 𝑋} ∈ 𝐸) → ((𝑈 ∈ (Vtx‘𝐺) ∧ 𝑊 ∈ (Vtx‘𝐺)) → (𝐺 ∈ FriendGraph → ((𝐺 ∈ USGraph ∧ (𝑈 ∈ 𝐷 ∧ 𝑊 ∈ 𝐷)) → (({𝑈, 𝐴} ∈ 𝐸 ∧ {𝑊, 𝐴} ∈ 𝐸) → 𝐴 = 𝑋))))))
89	88	com15 101	. . . . . 6 ⊢ ((𝐺 ∈ USGraph ∧ (𝑈 ∈ 𝐷 ∧ 𝑊 ∈ 𝐷)) → (({𝑈, 𝑋} ∈ 𝐸 ∧ {𝑊, 𝑋} ∈ 𝐸) → ((𝑈 ∈ (Vtx‘𝐺) ∧ 𝑊 ∈ (Vtx‘𝐺)) → (𝐺 ∈ FriendGraph → (𝑈 ≠ 𝑊 → (({𝑈, 𝐴} ∈ 𝐸 ∧ {𝑊, 𝐴} ∈ 𝐸) → 𝐴 = 𝑋))))))
90	13, 20, 89	mp2d 49	. . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ (𝑈 ∈ 𝐷 ∧ 𝑊 ∈ 𝐷)) → (𝐺 ∈ FriendGraph → (𝑈 ≠ 𝑊 → (({𝑈, 𝐴} ∈ 𝐸 ∧ {𝑊, 𝐴} ∈ 𝐸) → 𝐴 = 𝑋))))
91	90	ex 412	. . . 4 ⊢ (𝐺 ∈ USGraph → ((𝑈 ∈ 𝐷 ∧ 𝑊 ∈ 𝐷) → (𝐺 ∈ FriendGraph → (𝑈 ≠ 𝑊 → (({𝑈, 𝐴} ∈ 𝐸 ∧ {𝑊, 𝐴} ∈ 𝐸) → 𝐴 = 𝑋)))))
92	91	com23 86	. . 3 ⊢ (𝐺 ∈ USGraph → (𝐺 ∈ FriendGraph → ((𝑈 ∈ 𝐷 ∧ 𝑊 ∈ 𝐷) → (𝑈 ≠ 𝑊 → (({𝑈, 𝐴} ∈ 𝐸 ∧ {𝑊, 𝐴} ∈ 𝐸) → 𝐴 = 𝑋)))))
93	1, 92	mpcom 38	. 2 ⊢ (𝐺 ∈ FriendGraph → ((𝑈 ∈ 𝐷 ∧ 𝑊 ∈ 𝐷) → (𝑈 ≠ 𝑊 → (({𝑈, 𝐴} ∈ 𝐸 ∧ {𝑊, 𝐴} ∈ 𝐸) → 𝐴 = 𝑋))))
94	93	3imp 1110	1 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝑈 ∈ 𝐷 ∧ 𝑊 ∈ 𝐷) ∧ 𝑈 ≠ 𝑊) → (({𝑈, 𝐴} ∈ 𝐸 ∧ {𝑊, 𝐴} ∈ 𝐸) → 𝐴 = 𝑋))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∉ wnel 3029 {cpr 4579 ‘cfv 6482 (class class class)co 7349 Vtxcvtx 28945 Edgcedg 28996 USGraphcusgr 29098 NeighbVtx cnbgr 29281 FriendGraph cfrgr 30206

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-int 4897 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7800 df-1st 7924 df-2nd 7925 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-2o 8389 df-oadd 8392 df-er 8625 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-dju 9797 df-card 9835 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-nn 12129 df-2 12191 df-n0 12385 df-xnn0 12458 df-z 12472 df-uz 12736 df-fz 13411 df-hash 14238 df-edg 28997 df-upgr 29031 df-umgr 29032 df-usgr 29100 df-nbgr 29282 df-frgr 30207

This theorem is referenced by: frgrncvvdeqlem8 30254

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