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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 = {𝑥𝑉 ∣ ((𝑉 VDeg 𝐸)‘𝑥) = 𝐾}
frgrawopreg.b 𝐵 = (𝑉𝐴)
Assertion
Ref Expression
frgrawopreglem5 ((𝑉 FriendGrph 𝐸 ∧ 1 < (#‘𝐴) ∧ 1 < (#‘𝐵)) → ∃𝑎𝐴𝑥𝐴𝑏𝐵𝑦𝐵 ((𝑏𝑦𝑎𝑥) ∧ ({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.
StepHypRef Expression
1 frgrawopreg.a . . . 4 𝐴 = {𝑥𝑉 ∣ ((𝑉 VDeg 𝐸)‘𝑥) = 𝐾}
2 frgrawopreg.b . . . 4 𝐵 = (𝑉𝐴)
31, 2frgrawopreglem1 26571 . . 3 (𝑉 FriendGrph 𝐸 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
4 hashgt12el 13070 . . . . . . . . 9 ((𝐴 ∈ V ∧ 1 < (#‘𝐴)) → ∃𝑎𝐴𝑥𝐴 𝑎𝑥)
54ex 449 . . . . . . . 8 (𝐴 ∈ V → (1 < (#‘𝐴) → ∃𝑎𝐴𝑥𝐴 𝑎𝑥))
65ad2antrr 758 . . . . . . 7 (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 1 < (#‘𝐵)) → (1 < (#‘𝐴) → ∃𝑎𝐴𝑥𝐴 𝑎𝑥))
76imp 444 . . . . . 6 ((((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 1 < (#‘𝐵)) ∧ 1 < (#‘𝐴)) → ∃𝑎𝐴𝑥𝐴 𝑎𝑥)
8 hashgt12el 13070 . . . . . . . 8 ((𝐵 ∈ V ∧ 1 < (#‘𝐵)) → ∃𝑏𝐵𝑦𝐵 𝑏𝑦)
98adantll 746 . . . . . . 7 (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 1 < (#‘𝐵)) → ∃𝑏𝐵𝑦𝐵 𝑏𝑦)
109adantr 480 . . . . . 6 ((((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 1 < (#‘𝐵)) ∧ 1 < (#‘𝐴)) → ∃𝑏𝐵𝑦𝐵 𝑏𝑦)
11 reeanv 3086 . . . . . . 7 (∃𝑎𝐴𝑏𝐵 (∃𝑥𝐴 𝑎𝑥 ∧ ∃𝑦𝐵 𝑏𝑦) ↔ (∃𝑎𝐴𝑥𝐴 𝑎𝑥 ∧ ∃𝑏𝐵𝑦𝐵 𝑏𝑦))
12 reeanv 3086 . . . . . . . . 9 (∃𝑥𝐴𝑦𝐵 (𝑎𝑥𝑏𝑦) ↔ (∃𝑥𝐴 𝑎𝑥 ∧ ∃𝑦𝐵 𝑏𝑦))
13122rexbii 3024 . . . . . . . 8 (∃𝑎𝐴𝑏𝐵𝑥𝐴𝑦𝐵 (𝑎𝑥𝑏𝑦) ↔ ∃𝑎𝐴𝑏𝐵 (∃𝑥𝐴 𝑎𝑥 ∧ ∃𝑦𝐵 𝑏𝑦))
14 rexcom 3080 . . . . . . . . . . 11 (∃𝑏𝐵𝑥𝐴𝑦𝐵 (𝑎𝑥𝑏𝑦) ↔ ∃𝑥𝐴𝑏𝐵𝑦𝐵 (𝑎𝑥𝑏𝑦))
15 simpr 476 . . . . . . . . . . . . . . . . 17 ((((((𝑉 FriendGrph 𝐸𝑎𝐴) ∧ 𝑥𝐴) ∧ 𝑏𝐵) ∧ 𝑦𝐵) ∧ (𝑎𝑥𝑏𝑦)) → (𝑎𝑥𝑏𝑦))
1615ancomd 466 . . . . . . . . . . . . . . . 16 ((((((𝑉 FriendGrph 𝐸𝑎𝐴) ∧ 𝑥𝐴) ∧ 𝑏𝐵) ∧ 𝑦𝐵) ∧ (𝑎𝑥𝑏𝑦)) → (𝑏𝑦𝑎𝑥))
171, 2frgrawopreglem4 26574 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑉 FriendGrph 𝐸 → ∀𝑎𝐴𝑏𝐵 {𝑎, 𝑏} ∈ ran 𝐸)
18 rsp2 2920 . . . . . . . . . . . . . . . . . . . . . . 23 (∀𝑎𝐴𝑏𝐵 {𝑎, 𝑏} ∈ ran 𝐸 → ((𝑎𝐴𝑏𝐵) → {𝑎, 𝑏} ∈ ran 𝐸))
1917, 18syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝑉 FriendGrph 𝐸 → ((𝑎𝐴𝑏𝐵) → {𝑎, 𝑏} ∈ ran 𝐸))
2019expdimp 452 . . . . . . . . . . . . . . . . . . . . 21 ((𝑉 FriendGrph 𝐸𝑎𝐴) → (𝑏𝐵 → {𝑎, 𝑏} ∈ ran 𝐸))
2120adantr 480 . . . . . . . . . . . . . . . . . . . 20 (((𝑉 FriendGrph 𝐸𝑎𝐴) ∧ 𝑥𝐴) → (𝑏𝐵 → {𝑎, 𝑏} ∈ ran 𝐸))
2221imp 444 . . . . . . . . . . . . . . . . . . 19 ((((𝑉 FriendGrph 𝐸𝑎𝐴) ∧ 𝑥𝐴) ∧ 𝑏𝐵) → {𝑎, 𝑏} ∈ ran 𝐸)
2322adantr 480 . . . . . . . . . . . . . . . . . 18 (((((𝑉 FriendGrph 𝐸𝑎𝐴) ∧ 𝑥𝐴) ∧ 𝑏𝐵) ∧ 𝑦𝐵) → {𝑎, 𝑏} ∈ ran 𝐸)
241, 2frgrawopreglem4 26574 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑉 FriendGrph 𝐸 → ∀𝑧𝐴𝑐𝐵 {𝑧, 𝑐} ∈ ran 𝐸)
25 preq1 4212 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑧 = 𝑥 → {𝑧, 𝑐} = {𝑥, 𝑐})
2625eleq1d 2672 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧 = 𝑥 → ({𝑧, 𝑐} ∈ ran 𝐸 ↔ {𝑥, 𝑐} ∈ ran 𝐸))
27 preq2 4213 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑐 = 𝑏 → {𝑥, 𝑐} = {𝑥, 𝑏})
2827eleq1d 2672 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑐 = 𝑏 → ({𝑥, 𝑐} ∈ ran 𝐸 ↔ {𝑥, 𝑏} ∈ ran 𝐸))
2926, 28cbvral2v 3155 . . . . . . . . . . . . . . . . . . . . . . . 24 (∀𝑧𝐴𝑐𝐵 {𝑧, 𝑐} ∈ ran 𝐸 ↔ ∀𝑥𝐴𝑏𝐵 {𝑥, 𝑏} ∈ ran 𝐸)
30 rsp2 2920 . . . . . . . . . . . . . . . . . . . . . . . 24 (∀𝑥𝐴𝑏𝐵 {𝑥, 𝑏} ∈ ran 𝐸 → ((𝑥𝐴𝑏𝐵) → {𝑥, 𝑏} ∈ ran 𝐸))
3129, 30sylbi 206 . . . . . . . . . . . . . . . . . . . . . . 23 (∀𝑧𝐴𝑐𝐵 {𝑧, 𝑐} ∈ ran 𝐸 → ((𝑥𝐴𝑏𝐵) → {𝑥, 𝑏} ∈ ran 𝐸))
3224, 31syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝑉 FriendGrph 𝐸 → ((𝑥𝐴𝑏𝐵) → {𝑥, 𝑏} ∈ ran 𝐸))
3332expd 451 . . . . . . . . . . . . . . . . . . . . 21 (𝑉 FriendGrph 𝐸 → (𝑥𝐴 → (𝑏𝐵 → {𝑥, 𝑏} ∈ ran 𝐸)))
3433adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝑉 FriendGrph 𝐸𝑎𝐴) → (𝑥𝐴 → (𝑏𝐵 → {𝑥, 𝑏} ∈ ran 𝐸)))
3534imp31 447 . . . . . . . . . . . . . . . . . . 19 ((((𝑉 FriendGrph 𝐸𝑎𝐴) ∧ 𝑥𝐴) ∧ 𝑏𝐵) → {𝑥, 𝑏} ∈ ran 𝐸)
3635adantr 480 . . . . . . . . . . . . . . . . . 18 (((((𝑉 FriendGrph 𝐸𝑎𝐴) ∧ 𝑥𝐴) ∧ 𝑏𝐵) ∧ 𝑦𝐵) → {𝑥, 𝑏} ∈ ran 𝐸)
3723, 36jca 553 . . . . . . . . . . . . . . . . 17 (((((𝑉 FriendGrph 𝐸𝑎𝐴) ∧ 𝑥𝐴) ∧ 𝑏𝐵) ∧ 𝑦𝐵) → ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸))
3837adantr 480 . . . . . . . . . . . . . . . 16 ((((((𝑉 FriendGrph 𝐸𝑎𝐴) ∧ 𝑥𝐴) ∧ 𝑏𝐵) ∧ 𝑦𝐵) ∧ (𝑎𝑥𝑏𝑦)) → ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸))
391, 2frgrawopreglem4 26574 . . . . . . . . . . . . . . . . . . . . . 22 (𝑉 FriendGrph 𝐸 → ∀𝑎𝐴𝑦𝐵 {𝑎, 𝑦} ∈ ran 𝐸)
40 rsp2 2920 . . . . . . . . . . . . . . . . . . . . . 22 (∀𝑎𝐴𝑦𝐵 {𝑎, 𝑦} ∈ ran 𝐸 → ((𝑎𝐴𝑦𝐵) → {𝑎, 𝑦} ∈ ran 𝐸))
4139, 40syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝑉 FriendGrph 𝐸 → ((𝑎𝐴𝑦𝐵) → {𝑎, 𝑦} ∈ ran 𝐸))
4241expdimp 452 . . . . . . . . . . . . . . . . . . . 20 ((𝑉 FriendGrph 𝐸𝑎𝐴) → (𝑦𝐵 → {𝑎, 𝑦} ∈ ran 𝐸))
4342ad2antrr 758 . . . . . . . . . . . . . . . . . . 19 ((((𝑉 FriendGrph 𝐸𝑎𝐴) ∧ 𝑥𝐴) ∧ 𝑏𝐵) → (𝑦𝐵 → {𝑎, 𝑦} ∈ ran 𝐸))
4443imp 444 . . . . . . . . . . . . . . . . . 18 (((((𝑉 FriendGrph 𝐸𝑎𝐴) ∧ 𝑥𝐴) ∧ 𝑏𝐵) ∧ 𝑦𝐵) → {𝑎, 𝑦} ∈ ran 𝐸)
45 preq2 4213 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑐 = 𝑦 → {𝑥, 𝑐} = {𝑥, 𝑦})
4645eleq1d 2672 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑐 = 𝑦 → ({𝑥, 𝑐} ∈ ran 𝐸 ↔ {𝑥, 𝑦} ∈ ran 𝐸))
4726, 46cbvral2v 3155 . . . . . . . . . . . . . . . . . . . . . . . . 25 (∀𝑧𝐴𝑐𝐵 {𝑧, 𝑐} ∈ ran 𝐸 ↔ ∀𝑥𝐴𝑦𝐵 {𝑥, 𝑦} ∈ ran 𝐸)
48 rsp2 2920 . . . . . . . . . . . . . . . . . . . . . . . . 25 (∀𝑥𝐴𝑦𝐵 {𝑥, 𝑦} ∈ ran 𝐸 → ((𝑥𝐴𝑦𝐵) → {𝑥, 𝑦} ∈ ran 𝐸))
4947, 48sylbi 206 . . . . . . . . . . . . . . . . . . . . . . . 24 (∀𝑧𝐴𝑐𝐵 {𝑧, 𝑐} ∈ ran 𝐸 → ((𝑥𝐴𝑦𝐵) → {𝑥, 𝑦} ∈ ran 𝐸))
5024, 49syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑉 FriendGrph 𝐸 → ((𝑥𝐴𝑦𝐵) → {𝑥, 𝑦} ∈ ran 𝐸))
5150expd 451 . . . . . . . . . . . . . . . . . . . . . 22 (𝑉 FriendGrph 𝐸 → (𝑥𝐴 → (𝑦𝐵 → {𝑥, 𝑦} ∈ ran 𝐸)))
5251adantr 480 . . . . . . . . . . . . . . . . . . . . 21 ((𝑉 FriendGrph 𝐸𝑎𝐴) → (𝑥𝐴 → (𝑦𝐵 → {𝑥, 𝑦} ∈ ran 𝐸)))
5352imp 444 . . . . . . . . . . . . . . . . . . . 20 (((𝑉 FriendGrph 𝐸𝑎𝐴) ∧ 𝑥𝐴) → (𝑦𝐵 → {𝑥, 𝑦} ∈ ran 𝐸))
5453adantr 480 . . . . . . . . . . . . . . . . . . 19 ((((𝑉 FriendGrph 𝐸𝑎𝐴) ∧ 𝑥𝐴) ∧ 𝑏𝐵) → (𝑦𝐵 → {𝑥, 𝑦} ∈ ran 𝐸))
5554imp 444 . . . . . . . . . . . . . . . . . 18 (((((𝑉 FriendGrph 𝐸𝑎𝐴) ∧ 𝑥𝐴) ∧ 𝑏𝐵) ∧ 𝑦𝐵) → {𝑥, 𝑦} ∈ ran 𝐸)
5644, 55jca 553 . . . . . . . . . . . . . . . . 17 (((((𝑉 FriendGrph 𝐸𝑎𝐴) ∧ 𝑥𝐴) ∧ 𝑏𝐵) ∧ 𝑦𝐵) → ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))
5756adantr 480 . . . . . . . . . . . . . . . 16 ((((((𝑉 FriendGrph 𝐸𝑎𝐴) ∧ 𝑥𝐴) ∧ 𝑏𝐵) ∧ 𝑦𝐵) ∧ (𝑎𝑥𝑏𝑦)) → ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))
5816, 38, 573jca 1235 . . . . . . . . . . . . . . 15 ((((((𝑉 FriendGrph 𝐸𝑎𝐴) ∧ 𝑥𝐴) ∧ 𝑏𝐵) ∧ 𝑦𝐵) ∧ (𝑎𝑥𝑏𝑦)) → ((𝑏𝑦𝑎𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸)))
5958ex 449 . . . . . . . . . . . . . 14 (((((𝑉 FriendGrph 𝐸𝑎𝐴) ∧ 𝑥𝐴) ∧ 𝑏𝐵) ∧ 𝑦𝐵) → ((𝑎𝑥𝑏𝑦) → ((𝑏𝑦𝑎𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))))
6059reximdva 3000 . . . . . . . . . . . . 13 ((((𝑉 FriendGrph 𝐸𝑎𝐴) ∧ 𝑥𝐴) ∧ 𝑏𝐵) → (∃𝑦𝐵 (𝑎𝑥𝑏𝑦) → ∃𝑦𝐵 ((𝑏𝑦𝑎𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))))
6160reximdva 3000 . . . . . . . . . . . 12 (((𝑉 FriendGrph 𝐸𝑎𝐴) ∧ 𝑥𝐴) → (∃𝑏𝐵𝑦𝐵 (𝑎𝑥𝑏𝑦) → ∃𝑏𝐵𝑦𝐵 ((𝑏𝑦𝑎𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))))
6261reximdva 3000 . . . . . . . . . . 11 ((𝑉 FriendGrph 𝐸𝑎𝐴) → (∃𝑥𝐴𝑏𝐵𝑦𝐵 (𝑎𝑥𝑏𝑦) → ∃𝑥𝐴𝑏𝐵𝑦𝐵 ((𝑏𝑦𝑎𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))))
6314, 62syl5bi 231 . . . . . . . . . 10 ((𝑉 FriendGrph 𝐸𝑎𝐴) → (∃𝑏𝐵𝑥𝐴𝑦𝐵 (𝑎𝑥𝑏𝑦) → ∃𝑥𝐴𝑏𝐵𝑦𝐵 ((𝑏𝑦𝑎𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))))
6463reximdva 3000 . . . . . . . . 9 (𝑉 FriendGrph 𝐸 → (∃𝑎𝐴𝑏𝐵𝑥𝐴𝑦𝐵 (𝑎𝑥𝑏𝑦) → ∃𝑎𝐴𝑥𝐴𝑏𝐵𝑦𝐵 ((𝑏𝑦𝑎𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))))
6564com12 32 . . . . . . . 8 (∃𝑎𝐴𝑏𝐵𝑥𝐴𝑦𝐵 (𝑎𝑥𝑏𝑦) → (𝑉 FriendGrph 𝐸 → ∃𝑎𝐴𝑥𝐴𝑏𝐵𝑦𝐵 ((𝑏𝑦𝑎𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))))
6613, 65sylbir 224 . . . . . . 7 (∃𝑎𝐴𝑏𝐵 (∃𝑥𝐴 𝑎𝑥 ∧ ∃𝑦𝐵 𝑏𝑦) → (𝑉 FriendGrph 𝐸 → ∃𝑎𝐴𝑥𝐴𝑏𝐵𝑦𝐵 ((𝑏𝑦𝑎𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))))
6711, 66sylbir 224 . . . . . 6 ((∃𝑎𝐴𝑥𝐴 𝑎𝑥 ∧ ∃𝑏𝐵𝑦𝐵 𝑏𝑦) → (𝑉 FriendGrph 𝐸 → ∃𝑎𝐴𝑥𝐴𝑏𝐵𝑦𝐵 ((𝑏𝑦𝑎𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))))
687, 10, 67syl2anc 691 . . . . 5 ((((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 1 < (#‘𝐵)) ∧ 1 < (#‘𝐴)) → (𝑉 FriendGrph 𝐸 → ∃𝑎𝐴𝑥𝐴𝑏𝐵𝑦𝐵 ((𝑏𝑦𝑎𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))))
6968exp31 628 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (1 < (#‘𝐵) → (1 < (#‘𝐴) → (𝑉 FriendGrph 𝐸 → ∃𝑎𝐴𝑥𝐴𝑏𝐵𝑦𝐵 ((𝑏𝑦𝑎𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))))))
7069com24 93 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑉 FriendGrph 𝐸 → (1 < (#‘𝐴) → (1 < (#‘𝐵) → ∃𝑎𝐴𝑥𝐴𝑏𝐵𝑦𝐵 ((𝑏𝑦𝑎𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))))))
713, 70mpcom 37 . 2 (𝑉 FriendGrph 𝐸 → (1 < (#‘𝐴) → (1 < (#‘𝐵) → ∃𝑎𝐴𝑥𝐴𝑏𝐵𝑦𝐵 ((𝑏𝑦𝑎𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸)))))
72713imp 1249 1 ((𝑉 FriendGrph 𝐸 ∧ 1 < (#‘𝐴) ∧ 1 < (#‘𝐵)) → ∃𝑎𝐴𝑥𝐴𝑏𝐵𝑦𝐵 ((𝑏𝑦𝑎𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸)))
 Colors of variables: wff setvar class 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
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