frgrwopreglem5 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > frgrwopreglem5		Structured version Visualization version GIF version

Theorem frgrwopreglem5 29563

Description: Lemma 5 for frgrwopreg 29565. If 𝐴 as well as 𝐵 contain at least two vertices, there is a 4-cycle in a friendship graph. This corresponds to statement 6 in [Huneke] p. 2: "... otherwise, there are two different vertices in A, and they have two common neighbors in B, ...". (Contributed by Alexander van der Vekens, 31-Dec-2017.) (Proof shortened by AV, 5-Feb-2022.)

**Hypotheses**
Ref	Expression
frgrwopreg.v	⊢ 𝑉 = (Vtx‘𝐺)
frgrwopreg.d	⊢ 𝐷 = (VtxDeg‘𝐺)
frgrwopreg.a	⊢ 𝐴 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾}
frgrwopreg.b	⊢ 𝐵 = (𝑉 ∖ 𝐴)
frgrwopreg.e	⊢ 𝐸 = (Edg‘𝐺)

**Assertion**
Ref	Expression
frgrwopreglem5	⊢ ((𝐺 ∈ FriendGraph ∧ 1 < (♯‘𝐴) ∧ 1 < (♯‘𝐵)) → ∃𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦) ∧ ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑥} ∈ 𝐸) ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑎} ∈ 𝐸)))

Distinct variable groups: 𝑥,𝑉 𝑥,𝐴 𝑥,𝐺 𝑥,𝐾 𝑥,𝐷 𝐴,𝑏 𝑥,𝐵 𝑦,𝐷 𝐺,𝑎,𝑏,𝑦,𝑥 𝑦,𝑉 𝐴,𝑎,𝑦 𝐵,𝑎,𝑏,𝑦 𝑥,𝐸 Allowed substitution hints: 𝐷(𝑎,𝑏) 𝐸(𝑦,𝑎,𝑏) 𝐾(𝑦,𝑎,𝑏) 𝑉(𝑎,𝑏)

**Proof of Theorem frgrwopreglem5**
Step	Hyp	Ref	Expression
1		simpllr 774	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ FriendGraph ∧ 𝑎 ≠ 𝑥) ∧ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) ∧ (𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑎 ≠ 𝑥)
2	1	anim1i 615	. . . . . . . . . . . . 13 ⊢ (((((𝐺 ∈ FriendGraph ∧ 𝑎 ≠ 𝑥) ∧ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) ∧ (𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑏 ≠ 𝑦) → (𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦))
3		simplll 773	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐺 ∈ FriendGraph ∧ 𝑎 ≠ 𝑥) ∧ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) ∧ (𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐺 ∈ FriendGraph )
4		fveqeq2 6897	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑎 → ((𝐷‘𝑥) = 𝐾 ↔ (𝐷‘𝑎) = 𝐾))
5		frgrwopreg.a	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝐴 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾}
6	4, 5	elrab2 3685	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 ∈ 𝐴 ↔ (𝑎 ∈ 𝑉 ∧ (𝐷‘𝑎) = 𝐾))
7	6	simplbi 498	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 ∈ 𝐴 → 𝑎 ∈ 𝑉)
8		rabidim1 3453	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} → 𝑥 ∈ 𝑉)
9	8, 5	eleq2s 2851	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑉)
10	7, 9	anim12i 613	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑎 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉))
11	10	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑎 ≠ 𝑥) ∧ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → (𝑎 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉))
12		eldifi 4125	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 ∈ (𝑉 ∖ 𝐴) → 𝑏 ∈ 𝑉)
13		frgrwopreg.b	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐵 = (𝑉 ∖ 𝐴)
14	12, 13	eleq2s 2851	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 ∈ 𝐵 → 𝑏 ∈ 𝑉)
15		eldifi 4125	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ (𝑉 ∖ 𝐴) → 𝑦 ∈ 𝑉)
16	15, 13	eleq2s 2851	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝑉)
17	14, 16	anim12i 613	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑏 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉))
18	11, 17	anim12i 613	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐺 ∈ FriendGraph ∧ 𝑎 ≠ 𝑥) ∧ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) ∧ (𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑎 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)))
19		frgrwopreg.v	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑉 = (Vtx‘𝐺)
20		frgrwopreg.d	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐷 = (VtxDeg‘𝐺)
21		frgrwopreg.e	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐸 = (Edg‘𝐺)
22	19, 20, 5, 13, 21	frgrwopreglem5lem 29562	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝐷‘𝑎) = (𝐷‘𝑥) ∧ (𝐷‘𝑎) ≠ (𝐷‘𝑏) ∧ (𝐷‘𝑥) ≠ (𝐷‘𝑦)))
23	22	adantll 712	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐺 ∈ FriendGraph ∧ 𝑎 ≠ 𝑥) ∧ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) ∧ (𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝐷‘𝑎) = (𝐷‘𝑥) ∧ (𝐷‘𝑎) ≠ (𝐷‘𝑏) ∧ (𝐷‘𝑥) ≠ (𝐷‘𝑦)))
24	3, 18, 23	3jca 1128	. . . . . . . . . . . . . . 15 ⊢ ((((𝐺 ∈ FriendGraph ∧ 𝑎 ≠ 𝑥) ∧ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) ∧ (𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐺 ∈ FriendGraph ∧ ((𝑎 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ ((𝐷‘𝑎) = (𝐷‘𝑥) ∧ (𝐷‘𝑎) ≠ (𝐷‘𝑏) ∧ (𝐷‘𝑥) ≠ (𝐷‘𝑦))))
25	24	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((((𝐺 ∈ FriendGraph ∧ 𝑎 ≠ 𝑥) ∧ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) ∧ (𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑏 ≠ 𝑦) → (𝐺 ∈ FriendGraph ∧ ((𝑎 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ ((𝐷‘𝑎) = (𝐷‘𝑥) ∧ (𝐷‘𝑎) ≠ (𝐷‘𝑏) ∧ (𝐷‘𝑥) ≠ (𝐷‘𝑦))))
26	19, 20, 21	frgrwopreglem5a 29553	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ FriendGraph ∧ ((𝑎 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ ((𝐷‘𝑎) = (𝐷‘𝑥) ∧ (𝐷‘𝑎) ≠ (𝐷‘𝑏) ∧ (𝐷‘𝑥) ≠ (𝐷‘𝑦))) → (({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑥} ∈ 𝐸) ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑎} ∈ 𝐸)))
27	25, 26	syl 17	. . . . . . . . . . . . 13 ⊢ (((((𝐺 ∈ FriendGraph ∧ 𝑎 ≠ 𝑥) ∧ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) ∧ (𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑏 ≠ 𝑦) → (({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑥} ∈ 𝐸) ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑎} ∈ 𝐸)))
28		3anass 1095	. . . . . . . . . . . . 13 ⊢ (((𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦) ∧ ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑥} ∈ 𝐸) ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑎} ∈ 𝐸)) ↔ ((𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦) ∧ (({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑥} ∈ 𝐸) ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑎} ∈ 𝐸))))
29	2, 27, 28	sylanbrc 583	. . . . . . . . . . . 12 ⊢ (((((𝐺 ∈ FriendGraph ∧ 𝑎 ≠ 𝑥) ∧ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) ∧ (𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑏 ≠ 𝑦) → ((𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦) ∧ ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑥} ∈ 𝐸) ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑎} ∈ 𝐸)))
30	29	ex 413	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ FriendGraph ∧ 𝑎 ≠ 𝑥) ∧ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) ∧ (𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑏 ≠ 𝑦 → ((𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦) ∧ ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑥} ∈ 𝐸) ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑎} ∈ 𝐸))))
31	30	reximdvva 3205	. . . . . . . . . 10 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑎 ≠ 𝑥) ∧ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → (∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 𝑏 ≠ 𝑦 → ∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦) ∧ ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑥} ∈ 𝐸) ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑎} ∈ 𝐸))))
32	31	exp31 420	. . . . . . . . 9 ⊢ (𝐺 ∈ FriendGraph → (𝑎 ≠ 𝑥 → ((𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 𝑏 ≠ 𝑦 → ∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦) ∧ ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑥} ∈ 𝐸) ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑎} ∈ 𝐸))))))
33	32	com24 95	. . . . . . . 8 ⊢ (𝐺 ∈ FriendGraph → (∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 𝑏 ≠ 𝑦 → ((𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑎 ≠ 𝑥 → ∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦) ∧ ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑥} ∈ 𝐸) ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑎} ∈ 𝐸))))))
34	33	imp31 418	. . . . . . 7 ⊢ (((𝐺 ∈ FriendGraph ∧ ∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 𝑏 ≠ 𝑦) ∧ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → (𝑎 ≠ 𝑥 → ∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦) ∧ ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑥} ∈ 𝐸) ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑎} ∈ 𝐸))))
35	34	reximdvva 3205	. . . . . 6 ⊢ ((𝐺 ∈ FriendGraph ∧ ∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 𝑏 ≠ 𝑦) → (∃𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝐴 𝑎 ≠ 𝑥 → ∃𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦) ∧ ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑥} ∈ 𝐸) ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑎} ∈ 𝐸))))
36	35	ex 413	. . . . 5 ⊢ (𝐺 ∈ FriendGraph → (∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 𝑏 ≠ 𝑦 → (∃𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝐴 𝑎 ≠ 𝑥 → ∃𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦) ∧ ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑥} ∈ 𝐸) ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑎} ∈ 𝐸)))))
37	36	com13 88	. . . 4 ⊢ (∃𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝐴 𝑎 ≠ 𝑥 → (∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 𝑏 ≠ 𝑦 → (𝐺 ∈ FriendGraph → ∃𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦) ∧ ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑥} ∈ 𝐸) ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑎} ∈ 𝐸)))))
38	37	imp 407	. . 3 ⊢ ((∃𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝐴 𝑎 ≠ 𝑥 ∧ ∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 𝑏 ≠ 𝑦) → (𝐺 ∈ FriendGraph → ∃𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦) ∧ ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑥} ∈ 𝐸) ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑎} ∈ 𝐸))))
39	19, 20, 5, 13	frgrwopreglem1 29554	. . . 4 ⊢ (𝐴 ∈ V ∧ 𝐵 ∈ V)
40		hashgt12el 14378	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ 1 < (♯‘𝐴)) → ∃𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝐴 𝑎 ≠ 𝑥)
41	40	ex 413	. . . . 5 ⊢ (𝐴 ∈ V → (1 < (♯‘𝐴) → ∃𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝐴 𝑎 ≠ 𝑥))
42		hashgt12el 14378	. . . . . 6 ⊢ ((𝐵 ∈ V ∧ 1 < (♯‘𝐵)) → ∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 𝑏 ≠ 𝑦)
43	42	ex 413	. . . . 5 ⊢ (𝐵 ∈ V → (1 < (♯‘𝐵) → ∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 𝑏 ≠ 𝑦))
44	41, 43	im2anan9 620	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((1 < (♯‘𝐴) ∧ 1 < (♯‘𝐵)) → (∃𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝐴 𝑎 ≠ 𝑥 ∧ ∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 𝑏 ≠ 𝑦)))
45	39, 44	ax-mp 5	. . 3 ⊢ ((1 < (♯‘𝐴) ∧ 1 < (♯‘𝐵)) → (∃𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝐴 𝑎 ≠ 𝑥 ∧ ∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 𝑏 ≠ 𝑦))
46	38, 45	syl11 33	. 2 ⊢ (𝐺 ∈ FriendGraph → ((1 < (♯‘𝐴) ∧ 1 < (♯‘𝐵)) → ∃𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦) ∧ ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑥} ∈ 𝐸) ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑎} ∈ 𝐸))))
47	46	3impib 1116	1 ⊢ ((𝐺 ∈ FriendGraph ∧ 1 < (♯‘𝐴) ∧ 1 < (♯‘𝐵)) → ∃𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦) ∧ ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑥} ∈ 𝐸) ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑎} ∈ 𝐸)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∃wrex 3070 {crab 3432 Vcvv 3474 ∖ cdif 3944 {cpr 4629 class class class wbr 5147 ‘cfv 6540 1c1 11107 < clt 11244 ♯chash 14286 Vtxcvtx 28245 Edgcedg 28296 VtxDegcvtxdg 28711 FriendGraph cfrgr 29500

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-oadd 8466 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-dju 9892 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-n0 12469 df-xnn0 12541 df-z 12555 df-uz 12819 df-xadd 13089 df-fz 13481 df-hash 14287 df-edg 28297 df-uhgr 28307 df-ushgr 28308 df-upgr 28331 df-umgr 28332 df-uspgr 28399 df-usgr 28400 df-nbgr 28579 df-vtxdg 28712 df-frgr 29501

This theorem is referenced by: frgrwopreg 29565

W3C validator