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Theorem frgrnbnb 30759

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 30727	. . 3 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
2		frgrnbnb.n	. . . . . . . . . 10 ⊢ 𝐷 = (𝐺 NeighbVtx 𝑋)
3	2	eleq2i 2854	. . . . . . . . 9 ⊢ (𝑈 ∈ 𝐷 ↔ 𝑈 ∈ (𝐺 NeighbVtx 𝑋))
4		frgrnbnb.e	. . . . . . . . . . 11 ⊢ 𝐸 = (Edg‘𝐺)
5	4	nbusgreledg 29799	. . . . . . . . . 10 ⊢ (𝐺 ∈ USGraph → (𝑈 ∈ (𝐺 NeighbVtx 𝑋) ↔ {𝑈, 𝑋} ∈ 𝐸))
6	5	biimpd 232	. . . . . . . . 9 ⊢ (𝐺 ∈ USGraph → (𝑈 ∈ (𝐺 NeighbVtx 𝑋) → {𝑈, 𝑋} ∈ 𝐸))
7	3, 6	biimtrid 245	. . . . . . . 8 ⊢ (𝐺 ∈ USGraph → (𝑈 ∈ 𝐷 → {𝑈, 𝑋} ∈ 𝐸))
8	2	eleq2i 2854	. . . . . . . . 9 ⊢ (𝑊 ∈ 𝐷 ↔ 𝑊 ∈ (𝐺 NeighbVtx 𝑋))
9	4	nbusgreledg 29799	. . . . . . . . . 10 ⊢ (𝐺 ∈ USGraph → (𝑊 ∈ (𝐺 NeighbVtx 𝑋) ↔ {𝑊, 𝑋} ∈ 𝐸))
10	9	biimpd 232	. . . . . . . . 9 ⊢ (𝐺 ∈ USGraph → (𝑊 ∈ (𝐺 NeighbVtx 𝑋) → {𝑊, 𝑋} ∈ 𝐸))
11	8, 10	biimtrid 245	. . . . . . . 8 ⊢ (𝐺 ∈ USGraph → (𝑊 ∈ 𝐷 → {𝑊, 𝑋} ∈ 𝐸))
12	7, 11	anim12d 621	. . . . . . 7 ⊢ (𝐺 ∈ USGraph → ((𝑈 ∈ 𝐷 ∧ 𝑊 ∈ 𝐷) → ({𝑈, 𝑋} ∈ 𝐸 ∧ {𝑊, 𝑋} ∈ 𝐸)))
13	12	imp 412	. . . . . 6 ⊢ ((𝐺 ∈ USGraph ∧ (𝑈 ∈ 𝐷 ∧ 𝑊 ∈ 𝐷)) → ({𝑈, 𝑋} ∈ 𝐸 ∧ {𝑊, 𝑋} ∈ 𝐸))
14		eqid 2762	. . . . . . . . . 10 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
15	14	nbgrisvtx 29787	. . . . . . . . 9 ⊢ (𝑈 ∈ (𝐺 NeighbVtx 𝑋) → 𝑈 ∈ (Vtx‘𝐺))
16	15, 2	eleq2s 2880	. . . . . . . 8 ⊢ (𝑈 ∈ 𝐷 → 𝑈 ∈ (Vtx‘𝐺))
17	14	nbgrisvtx 29787	. . . . . . . . 9 ⊢ (𝑊 ∈ (𝐺 NeighbVtx 𝑋) → 𝑊 ∈ (Vtx‘𝐺))
18	17, 2	eleq2s 2880	. . . . . . . 8 ⊢ (𝑊 ∈ 𝐷 → 𝑊 ∈ (Vtx‘𝐺))
19	16, 18	anim12i 625	. . . . . . 7 ⊢ ((𝑈 ∈ 𝐷 ∧ 𝑊 ∈ 𝐷) → (𝑈 ∈ (Vtx‘𝐺) ∧ 𝑊 ∈ (Vtx‘𝐺)))
20	19	adantl 487	. . . . . 6 ⊢ ((𝐺 ∈ USGraph ∧ (𝑈 ∈ 𝐷 ∧ 𝑊 ∈ 𝐷)) → (𝑈 ∈ (Vtx‘𝐺) ∧ 𝑊 ∈ (Vtx‘𝐺)))
21	4, 14	usgrpredgv 29643	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ USGraph ∧ {𝑈, 𝐴} ∈ 𝐸) → (𝑈 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)))
22	21	ad2ant2r 760	. . . . . . . . . . . 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 27	. . . . . . . . . . . . . . . . . . . . . . 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 27	. . . . . . . . . . . . . . . . . . . . . 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 779	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝐺 ∈ USGraph ∧ ((𝑋 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)) ∧ (𝑈 ∈ (Vtx‘𝐺) ∧ 𝑊 ∈ (Vtx‘𝐺)))) ∧ (𝐴 ≠ 𝑋 ∧ 𝑈 ≠ 𝑊)) → 𝐺 ∈ USGraph)
27		simprrr 794	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝐺 ∈ USGraph ∧ ((𝑋 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)) ∧ (𝑈 ∈ (Vtx‘𝐺) ∧ 𝑊 ∈ (Vtx‘𝐺)))) → 𝑊 ∈ (Vtx‘𝐺))
28	27	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝐺 ∈ USGraph ∧ ((𝑋 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)) ∧ (𝑈 ∈ (Vtx‘𝐺) ∧ 𝑊 ∈ (Vtx‘𝐺)))) ∧ (𝐴 ≠ 𝑋 ∧ 𝑈 ≠ 𝑊)) → 𝑊 ∈ (Vtx‘𝐺))
29		simprrl 793	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝐺 ∈ USGraph ∧ ((𝑋 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)) ∧ (𝑈 ∈ (Vtx‘𝐺) ∧ 𝑊 ∈ (Vtx‘𝐺)))) → 𝑈 ∈ (Vtx‘𝐺))
30	29	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝐺 ∈ USGraph ∧ ((𝑋 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)) ∧ (𝑈 ∈ (Vtx‘𝐺) ∧ 𝑊 ∈ (Vtx‘𝐺)))) ∧ (𝐴 ≠ 𝑋 ∧ 𝑈 ≠ 𝑊)) → 𝑈 ∈ (Vtx‘𝐺))
31		necom 3010	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑈 ≠ 𝑊 ↔ 𝑊 ≠ 𝑈)
32	31	biimpi 219	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑈 ≠ 𝑊 → 𝑊 ≠ 𝑈)
33	32	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝐴 ≠ 𝑋 ∧ 𝑈 ≠ 𝑊) → 𝑊 ≠ 𝑈)
34	33	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1146	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 791	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝐺 ∈ USGraph ∧ ((𝑋 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)) ∧ (𝑈 ∈ (Vtx‘𝐺) ∧ 𝑊 ∈ (Vtx‘𝐺)))) → 𝑋 ∈ (Vtx‘𝐺))
37	36	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝐺 ∈ USGraph ∧ ((𝑋 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)) ∧ (𝑈 ∈ (Vtx‘𝐺) ∧ 𝑊 ∈ (Vtx‘𝐺)))) ∧ (𝐴 ≠ 𝑋 ∧ 𝑈 ≠ 𝑊)) → 𝑋 ∈ (Vtx‘𝐺))
38		simprlr 792	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝐺 ∈ USGraph ∧ ((𝑋 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)) ∧ (𝑈 ∈ (Vtx‘𝐺) ∧ 𝑊 ∈ (Vtx‘𝐺)))) → 𝐴 ∈ (Vtx‘𝐺))
39	38	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝐺 ∈ USGraph ∧ ((𝑋 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)) ∧ (𝑈 ∈ (Vtx‘𝐺) ∧ 𝑊 ∈ (Vtx‘𝐺)))) ∧ (𝐴 ≠ 𝑋 ∧ 𝑈 ≠ 𝑊)) → 𝐴 ∈ (Vtx‘𝐺))
40		necom 3010	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝐴 ≠ 𝑋 ↔ 𝑋 ≠ 𝐴)
41	40	biimpi 219	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝐴 ≠ 𝑋 → 𝑋 ≠ 𝐴)
42	41	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝐴 ≠ 𝑋 ∧ 𝑈 ≠ 𝑊) → 𝑋 ≠ 𝐴)
43	42	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1146	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1146	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 765	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4696	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ {𝑈, 𝑋} = {𝑋, 𝑈}
48	47	eleq1i 2853	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ({𝑈, 𝑋} ∈ 𝐸 ↔ {𝑋, 𝑈} ∈ 𝐸)
49	48	biimpi 219	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ({𝑈, 𝑋} ∈ 𝐸 → {𝑋, 𝑈} ∈ 𝐸)
50	49	anim1ci 628	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (({𝑈, 𝑋} ∈ 𝐸 ∧ {𝑊, 𝑋} ∈ 𝐸) → ({𝑊, 𝑋} ∈ 𝐸 ∧ {𝑋, 𝑈} ∈ 𝐸))
51	50	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4696	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ {𝑊, 𝐴} = {𝐴, 𝑊}
53	52	eleq1i 2853	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ({𝑊, 𝐴} ∈ 𝐸 ↔ {𝐴, 𝑊} ∈ 𝐸)
54	53	biimpi 219	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ({𝑊, 𝐴} ∈ 𝐸 → {𝐴, 𝑊} ∈ 𝐸)
55	54	anim2i 629	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (({𝑈, 𝐴} ∈ 𝐸 ∧ {𝑊, 𝐴} ∈ 𝐸) → ({𝑈, 𝐴} ∈ 𝐸 ∧ {𝐴, 𝑊} ∈ 𝐸))
56	51, 55	anim12i 625	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 30758	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 66	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3064	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝐺 ∉ FriendGraph ↔ ¬ 𝐺 ∈ FriendGraph )
61	59, 60	sylib 221	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 122	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 418	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 87	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 440	. . . . . . . . . . . . . . . . . . . . . . . . . 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 100	. . . . . . . . . . . . . . . . . . . . . . . . 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 39	. . . . . . . . . . . . . . . . . . . . . . . 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 102	. . . . . . . . . . . . . . . . . . . . . . 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 418	. . . . . . . . . . . . . . . . . . . . . 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 3040	. . . . . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . . . . 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 89	. . . . . . . . . . . . . . . . . . 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 418	. . . . . . . . . . . . . . . . . 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 100	. . . . . . . . . . . . . . . . 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 418	. . . . . . . . . . . . . . . 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 29781	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑈 ∈ (𝐺 NeighbVtx 𝑋) → 𝑋 ∈ (Vtx‘𝐺))
77	76, 2	eleq2s 2880	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑈 ∈ 𝐷 → 𝑋 ∈ (Vtx‘𝐺))
78	77	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑈 ∈ 𝐷 ∧ 𝑊 ∈ 𝐷) → 𝑋 ∈ (Vtx‘𝐺))
79	78	adantl 487	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ∈ USGraph ∧ (𝑈 ∈ 𝐷 ∧ 𝑊 ∈ 𝐷)) → 𝑋 ∈ (Vtx‘𝐺))
80	75, 79	syl11 34	. . . . . . . . . . . . . . 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 92	. . . . . . . . . . . . . 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 416	. . . . . . . . . . . . 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 487	. . . . . . . . . . . 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 39	. . . . . . . . . . 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 418	. . . . . . . . . 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 100	. . . . . . . . 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 97	. . . . . . . 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 418	. . . . . . 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 102	. . . . . 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 50	. . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ (𝑈 ∈ 𝐷 ∧ 𝑊 ∈ 𝐷)) → (𝐺 ∈ FriendGraph → (𝑈 ≠ 𝑊 → (({𝑈, 𝐴} ∈ 𝐸 ∧ {𝑊, 𝐴} ∈ 𝐸) → 𝐴 = 𝑋))))
91	90	ex 418	. . . 4 ⊢ (𝐺 ∈ USGraph → ((𝑈 ∈ 𝐷 ∧ 𝑊 ∈ 𝐷) → (𝐺 ∈ FriendGraph → (𝑈 ≠ 𝑊 → (({𝑈, 𝐴} ∈ 𝐸 ∧ {𝑊, 𝐴} ∈ 𝐸) → 𝐴 = 𝑋)))))
92	91	com23 87	. . 3 ⊢ (𝐺 ∈ USGraph → (𝐺 ∈ FriendGraph → ((𝑈 ∈ 𝐷 ∧ 𝑊 ∈ 𝐷) → (𝑈 ≠ 𝑊 → (({𝑈, 𝐴} ∈ 𝐸 ∧ {𝑊, 𝐴} ∈ 𝐸) → 𝐴 = 𝑋)))))
93	1, 92	mpcom 39	. 2 ⊢ (𝐺 ∈ FriendGraph → ((𝑈 ∈ 𝐷 ∧ 𝑊 ∈ 𝐷) → (𝑈 ≠ 𝑊 → (({𝑈, 𝐴} ∈ 𝐸 ∧ {𝑊, 𝐴} ∈ 𝐸) → 𝐴 = 𝑋))))
94	93	3imp 1128	1 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝑈 ∈ 𝐷 ∧ 𝑊 ∈ 𝐷) ∧ 𝑈 ≠ 𝑊) → (({𝑈, 𝐴} ∈ 𝐸 ∧ {𝑊, 𝐴} ∈ 𝐸) → 𝐴 = 𝑋))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ≠ wne 2957 ∉ wnel 3063 {cpr 4589 ‘cfv 6537 (class class class)co 7416 Vtxcvtx 29439 Edgcedg 29490 USGraphcusgr 29595 NeighbVtx cnbgr 29778 FriendGraph cfrgr 30724

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7739 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 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 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7866 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8458 df-2o 8459 df-oadd 8462 df-er 8699 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-dju 9909 df-card 9947 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11470 df-neg 11471 df-nn 12261 df-2 12330 df-n0 12532 df-xnn0 12605 df-z 12619 df-uz 12891 df-fz 13564 df-hash 14397 df-edg 29491 df-upgr 29525 df-umgr 29526 df-usgr 29597 df-nbgr 29779 df-frgr 30725

This theorem is used by: frgrncvvdeqlem8 30772

W3C validator