Theorem frgrawopreglem5 26575

Description: Lemma 5 for frgrawopreg 26576. If A as well as B contain at least two vertices in a friendship graph, there is a 4-cycle in the 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.)

Hypotheses

Ref	Expression
frgrawopreg.a	⊢ 𝐴 = {𝑥 ∈ 𝑉 ∣ ((𝑉 VDeg 𝐸)‘𝑥) = 𝐾}
frgrawopreg.b	⊢ 𝐵 = (𝑉 ∖ 𝐴)

Assertion

Ref	Expression
frgrawopreglem5	⊢ ((𝑉 FriendGrph 𝐸 ∧ 1 < (#‘𝐴) ∧ 1 < (#‘𝐵)) → ∃𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑏 ≠ 𝑦 ∧ 𝑎 ≠ 𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐸   𝑥,𝐾   𝑥,𝑉   𝐴,𝑏   𝑥,𝑦,𝑎,𝑏,𝐸   𝑉,𝑎,𝑏,𝑦   𝐴,𝑎,𝑦   𝐵,𝑎,𝑏,𝑥,𝑦

Allowed substitution hints:   𝐾(𝑦,𝑎,𝑏)

Proof of Theorem frgrawopreglem5

Dummy variables 𝑧 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		frgrawopreg.a	. . . 4 ⊢ 𝐴 = {𝑥 ∈ 𝑉 ∣ ((𝑉 VDeg 𝐸)‘𝑥) = 𝐾}
2		frgrawopreg.b	. . . 4 ⊢ 𝐵 = (𝑉 ∖ 𝐴)
3	1, 2	frgrawopreglem1 26571	. . 3 ⊢ (𝑉 FriendGrph 𝐸 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
4		hashgt12el 13070	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ 1 < (#‘𝐴)) → ∃𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝐴 𝑎 ≠ 𝑥)
5	4	ex 449	. . . . . . . 8 ⊢ (𝐴 ∈ V → (1 < (#‘𝐴) → ∃𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝐴 𝑎 ≠ 𝑥))
6	5	ad2antrr 758	. . . . . . 7 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 1 < (#‘𝐵)) → (1 < (#‘𝐴) → ∃𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝐴 𝑎 ≠ 𝑥))
7	6	imp 444	. . . . . 6 ⊢ ((((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 1 < (#‘𝐵)) ∧ 1 < (#‘𝐴)) → ∃𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝐴 𝑎 ≠ 𝑥)
8		hashgt12el 13070	. . . . . . . 8 ⊢ ((𝐵 ∈ V ∧ 1 < (#‘𝐵)) → ∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 𝑏 ≠ 𝑦)
9	8	adantll 746	. . . . . . 7 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 1 < (#‘𝐵)) → ∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 𝑏 ≠ 𝑦)
10	9	adantr 480	. . . . . 6 ⊢ ((((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 1 < (#‘𝐵)) ∧ 1 < (#‘𝐴)) → ∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 𝑏 ≠ 𝑦)
11		reeanv 3086	. . . . . . 7 ⊢ (∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 (∃𝑥 ∈ 𝐴 𝑎 ≠ 𝑥 ∧ ∃𝑦 ∈ 𝐵 𝑏 ≠ 𝑦) ↔ (∃𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝐴 𝑎 ≠ 𝑥 ∧ ∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 𝑏 ≠ 𝑦))
12		reeanv 3086	. . . . . . . . 9 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦) ↔ (∃𝑥 ∈ 𝐴 𝑎 ≠ 𝑥 ∧ ∃𝑦 ∈ 𝐵 𝑏 ≠ 𝑦))
13	12	2rexbii 3024	. . . . . . . 8 ⊢ (∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦) ↔ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 (∃𝑥 ∈ 𝐴 𝑎 ≠ 𝑥 ∧ ∃𝑦 ∈ 𝐵 𝑏 ≠ 𝑦))
14		rexcom 3080	. . . . . . . . . . 11 ⊢ (∃𝑏 ∈ 𝐵 ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦) ↔ ∃𝑥 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 (𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦))
15		simpr 476	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝑉 FriendGrph 𝐸 ∧ 𝑎 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦)) → (𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦))
16	15	ancomd 466	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑉 FriendGrph 𝐸 ∧ 𝑎 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦)) → (𝑏 ≠ 𝑦 ∧ 𝑎 ≠ 𝑥))
17	1, 2	frgrawopreglem4 26574	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑉 FriendGrph 𝐸 → ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 {𝑎, 𝑏} ∈ ran 𝐸)
18		rsp2 2920	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 {𝑎, 𝑏} ∈ ran 𝐸 → ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → {𝑎, 𝑏} ∈ ran 𝐸))
19	17, 18	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑉 FriendGrph 𝐸 → ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → {𝑎, 𝑏} ∈ ran 𝐸))
20	19	expdimp 452	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝑎 ∈ 𝐴) → (𝑏 ∈ 𝐵 → {𝑎, 𝑏} ∈ ran 𝐸))
21	20	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑉 FriendGrph 𝐸 ∧ 𝑎 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → (𝑏 ∈ 𝐵 → {𝑎, 𝑏} ∈ ran 𝐸))
22	21	imp 444	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑉 FriendGrph 𝐸 ∧ 𝑎 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) ∧ 𝑏 ∈ 𝐵) → {𝑎, 𝑏} ∈ ran 𝐸)
23	22	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑉 FriendGrph 𝐸 ∧ 𝑎 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → {𝑎, 𝑏} ∈ ran 𝐸)
24	1, 2	frgrawopreglem4 26574	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑉 FriendGrph 𝐸 → ∀𝑧 ∈ 𝐴 ∀𝑐 ∈ 𝐵 {𝑧, 𝑐} ∈ ran 𝐸)
25		preq1 4212	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑧 = 𝑥 → {𝑧, 𝑐} = {𝑥, 𝑐})
26	25	eleq1d 2672	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑧 = 𝑥 → ({𝑧, 𝑐} ∈ ran 𝐸 ↔ {𝑥, 𝑐} ∈ ran 𝐸))
27		preq2 4213	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑐 = 𝑏 → {𝑥, 𝑐} = {𝑥, 𝑏})
28	27	eleq1d 2672	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑐 = 𝑏 → ({𝑥, 𝑐} ∈ ran 𝐸 ↔ {𝑥, 𝑏} ∈ ran 𝐸))
29	26, 28	cbvral2v 3155	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (∀𝑧 ∈ 𝐴 ∀𝑐 ∈ 𝐵 {𝑧, 𝑐} ∈ ran 𝐸 ↔ ∀𝑥 ∈ 𝐴 ∀𝑏 ∈ 𝐵 {𝑥, 𝑏} ∈ ran 𝐸)
30		rsp2 2920	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑏 ∈ 𝐵 {𝑥, 𝑏} ∈ ran 𝐸 → ((𝑥 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → {𝑥, 𝑏} ∈ ran 𝐸))
31	29, 30	sylbi 206	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∀𝑧 ∈ 𝐴 ∀𝑐 ∈ 𝐵 {𝑧, 𝑐} ∈ ran 𝐸 → ((𝑥 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → {𝑥, 𝑏} ∈ ran 𝐸))
32	24, 31	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑉 FriendGrph 𝐸 → ((𝑥 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → {𝑥, 𝑏} ∈ ran 𝐸))
33	32	expd 451	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑉 FriendGrph 𝐸 → (𝑥 ∈ 𝐴 → (𝑏 ∈ 𝐵 → {𝑥, 𝑏} ∈ ran 𝐸)))
34	33	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝑎 ∈ 𝐴) → (𝑥 ∈ 𝐴 → (𝑏 ∈ 𝐵 → {𝑥, 𝑏} ∈ ran 𝐸)))
35	34	imp31 447	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑉 FriendGrph 𝐸 ∧ 𝑎 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) ∧ 𝑏 ∈ 𝐵) → {𝑥, 𝑏} ∈ ran 𝐸)
36	35	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑉 FriendGrph 𝐸 ∧ 𝑎 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → {𝑥, 𝑏} ∈ ran 𝐸)
37	23, 36	jca 553	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑉 FriendGrph 𝐸 ∧ 𝑎 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸))
38	37	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑉 FriendGrph 𝐸 ∧ 𝑎 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦)) → ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸))
39	1, 2	frgrawopreglem4 26574	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑉 FriendGrph 𝐸 → ∀𝑎 ∈ 𝐴 ∀𝑦 ∈ 𝐵 {𝑎, 𝑦} ∈ ran 𝐸)
40		rsp2 2920	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∀𝑎 ∈ 𝐴 ∀𝑦 ∈ 𝐵 {𝑎, 𝑦} ∈ ran 𝐸 → ((𝑎 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → {𝑎, 𝑦} ∈ ran 𝐸))
41	39, 40	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑉 FriendGrph 𝐸 → ((𝑎 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → {𝑎, 𝑦} ∈ ran 𝐸))
42	41	expdimp 452	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝑎 ∈ 𝐴) → (𝑦 ∈ 𝐵 → {𝑎, 𝑦} ∈ ran 𝐸))
43	42	ad2antrr 758	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑉 FriendGrph 𝐸 ∧ 𝑎 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) ∧ 𝑏 ∈ 𝐵) → (𝑦 ∈ 𝐵 → {𝑎, 𝑦} ∈ ran 𝐸))
44	43	imp 444	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑉 FriendGrph 𝐸 ∧ 𝑎 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → {𝑎, 𝑦} ∈ ran 𝐸)
45		preq2 4213	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑐 = 𝑦 → {𝑥, 𝑐} = {𝑥, 𝑦})
46	45	eleq1d 2672	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑐 = 𝑦 → ({𝑥, 𝑐} ∈ ran 𝐸 ↔ {𝑥, 𝑦} ∈ ran 𝐸))
47	26, 46	cbvral2v 3155	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (∀𝑧 ∈ 𝐴 ∀𝑐 ∈ 𝐵 {𝑧, 𝑐} ∈ ran 𝐸 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 {𝑥, 𝑦} ∈ ran 𝐸)
48		rsp2 2920	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 {𝑥, 𝑦} ∈ ran 𝐸 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → {𝑥, 𝑦} ∈ ran 𝐸))
49	47, 48	sylbi 206	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (∀𝑧 ∈ 𝐴 ∀𝑐 ∈ 𝐵 {𝑧, 𝑐} ∈ ran 𝐸 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → {𝑥, 𝑦} ∈ ran 𝐸))
50	24, 49	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑉 FriendGrph 𝐸 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → {𝑥, 𝑦} ∈ ran 𝐸))
51	50	expd 451	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑉 FriendGrph 𝐸 → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → {𝑥, 𝑦} ∈ ran 𝐸)))
52	51	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝑎 ∈ 𝐴) → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → {𝑥, 𝑦} ∈ ran 𝐸)))
53	52	imp 444	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑉 FriendGrph 𝐸 ∧ 𝑎 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ 𝐵 → {𝑥, 𝑦} ∈ ran 𝐸))
54	53	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑉 FriendGrph 𝐸 ∧ 𝑎 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) ∧ 𝑏 ∈ 𝐵) → (𝑦 ∈ 𝐵 → {𝑥, 𝑦} ∈ ran 𝐸))
55	54	imp 444	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑉 FriendGrph 𝐸 ∧ 𝑎 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → {𝑥, 𝑦} ∈ ran 𝐸)
56	44, 55	jca 553	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑉 FriendGrph 𝐸 ∧ 𝑎 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))
57	56	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑉 FriendGrph 𝐸 ∧ 𝑎 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦)) → ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))
58	16, 38, 57	3jca 1235	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑉 FriendGrph 𝐸 ∧ 𝑎 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦)) → ((𝑏 ≠ 𝑦 ∧ 𝑎 ≠ 𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸)))
59	58	ex 449	. . . . . . . . . . . . . 14 ⊢ (((((𝑉 FriendGrph 𝐸 ∧ 𝑎 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → ((𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦) → ((𝑏 ≠ 𝑦 ∧ 𝑎 ≠ 𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))))
60	59	reximdva 3000	. . . . . . . . . . . . 13 ⊢ ((((𝑉 FriendGrph 𝐸 ∧ 𝑎 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) ∧ 𝑏 ∈ 𝐵) → (∃𝑦 ∈ 𝐵 (𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦) → ∃𝑦 ∈ 𝐵 ((𝑏 ≠ 𝑦 ∧ 𝑎 ≠ 𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))))
61	60	reximdva 3000	. . . . . . . . . . . 12 ⊢ (((𝑉 FriendGrph 𝐸 ∧ 𝑎 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → (∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 (𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦) → ∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑏 ≠ 𝑦 ∧ 𝑎 ≠ 𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))))
62	61	reximdva 3000	. . . . . . . . . . 11 ⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝑎 ∈ 𝐴) → (∃𝑥 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 (𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦) → ∃𝑥 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑏 ≠ 𝑦 ∧ 𝑎 ≠ 𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))))
63	14, 62	syl5bi 231	. . . . . . . . . 10 ⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝑎 ∈ 𝐴) → (∃𝑏 ∈ 𝐵 ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦) → ∃𝑥 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑏 ≠ 𝑦 ∧ 𝑎 ≠ 𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))))
64	63	reximdva 3000	. . . . . . . . 9 ⊢ (𝑉 FriendGrph 𝐸 → (∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦) → ∃𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑏 ≠ 𝑦 ∧ 𝑎 ≠ 𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))))
65	64	com12 32	. . . . . . . 8 ⊢ (∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦) → (𝑉 FriendGrph 𝐸 → ∃𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑏 ≠ 𝑦 ∧ 𝑎 ≠ 𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))))
66	13, 65	sylbir 224	. . . . . . 7 ⊢ (∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 (∃𝑥 ∈ 𝐴 𝑎 ≠ 𝑥 ∧ ∃𝑦 ∈ 𝐵 𝑏 ≠ 𝑦) → (𝑉 FriendGrph 𝐸 → ∃𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑏 ≠ 𝑦 ∧ 𝑎 ≠ 𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))))
67	11, 66	sylbir 224	. . . . . 6 ⊢ ((∃𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝐴 𝑎 ≠ 𝑥 ∧ ∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 𝑏 ≠ 𝑦) → (𝑉 FriendGrph 𝐸 → ∃𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑏 ≠ 𝑦 ∧ 𝑎 ≠ 𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))))
68	7, 10, 67	syl2anc 691	. . . . 5 ⊢ ((((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 1 < (#‘𝐵)) ∧ 1 < (#‘𝐴)) → (𝑉 FriendGrph 𝐸 → ∃𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑏 ≠ 𝑦 ∧ 𝑎 ≠ 𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))))
69	68	exp31 628	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (1 < (#‘𝐵) → (1 < (#‘𝐴) → (𝑉 FriendGrph 𝐸 → ∃𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑏 ≠ 𝑦 ∧ 𝑎 ≠ 𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))))))
70	69	com24 93	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑉 FriendGrph 𝐸 → (1 < (#‘𝐴) → (1 < (#‘𝐵) → ∃𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑏 ≠ 𝑦 ∧ 𝑎 ≠ 𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))))))
71	3, 70	mpcom 37	. 2 ⊢ (𝑉 FriendGrph 𝐸 → (1 < (#‘𝐴) → (1 < (#‘𝐵) → ∃𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑏 ≠ 𝑦 ∧ 𝑎 ≠ 𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸)))))
72	71	3imp 1249	1 ⊢ ((𝑉 FriendGrph 𝐸 ∧ 1 < (#‘𝐴) ∧ 1 < (#‘𝐵)) → ∃𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑏 ≠ 𝑦 ∧ 𝑎 ≠ 𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸)))

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173   ∖ cdif 3537  {cpr 4127   class class class wbr 4583  ran crn 5039  ‘cfv 5804  (class class class)co 6549  1c1 9816   < clt 9953  #chash 12979   VDeg cvdg 26420   FriendGrph cfrgra 26515

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-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  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-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  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-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-xadd 11823  df-fz 12198  df-hash 12980  df-usgra 25862  df-nbgra 25949  df-vdgr 26421  df-frgra 26516

This theorem is referenced by:  frgrawopreg  26576