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Theorem frgrwopreg 27872
Description: In a friendship graph there are either no vertices (𝐴 = ∅) or exactly one vertex ((♯‘𝐴) = 1) having degree 𝐾, or all (𝐵 = ∅) or all except one vertices ((♯‘𝐵) = 1) have degree 𝐾. (Contributed by Alexander van der Vekens, 31-Dec-2017.) (Revised by AV, 10-May-2021.) (Proof shortened by AV, 3-Jan-2022.)
Hypotheses
Ref Expression
frgrwopreg.v 𝑉 = (Vtx‘𝐺)
frgrwopreg.d 𝐷 = (VtxDeg‘𝐺)
frgrwopreg.a 𝐴 = {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}
frgrwopreg.b 𝐵 = (𝑉𝐴)
Assertion
Ref Expression
frgrwopreg (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))
Distinct variable groups:   𝑥,𝑉   𝑥,𝐴   𝑥,𝐺   𝑥,𝐾   𝑥,𝐷   𝑥,𝐵

Proof of Theorem frgrwopreg
Dummy variables 𝑎 𝑏 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frgrwopreg.v . . 3 𝑉 = (Vtx‘𝐺)
2 frgrwopreg.d . . 3 𝐷 = (VtxDeg‘𝐺)
3 frgrwopreg.a . . 3 𝐴 = {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}
4 frgrwopreg.b . . 3 𝐵 = (𝑉𝐴)
51, 2, 3, 4frgrwopreglem1 27861 . 2 (𝐴 ∈ V ∧ 𝐵 ∈ V)
6 hashv01gt1 13526 . . . 4 (𝐴 ∈ V → ((♯‘𝐴) = 0 ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴)))
7 hasheq0 13545 . . . . . 6 (𝐴 ∈ V → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅))
8 biidd 254 . . . . . 6 (𝐴 ∈ V → ((♯‘𝐴) = 1 ↔ (♯‘𝐴) = 1))
9 biidd 254 . . . . . 6 (𝐴 ∈ V → (1 < (♯‘𝐴) ↔ 1 < (♯‘𝐴)))
107, 8, 93orbi123d 1415 . . . . 5 (𝐴 ∈ V → (((♯‘𝐴) = 0 ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴)) ↔ (𝐴 = ∅ ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴))))
11 hashv01gt1 13526 . . . . . . 7 (𝐵 ∈ V → ((♯‘𝐵) = 0 ∨ (♯‘𝐵) = 1 ∨ 1 < (♯‘𝐵)))
12 hasheq0 13545 . . . . . . . . 9 (𝐵 ∈ V → ((♯‘𝐵) = 0 ↔ 𝐵 = ∅))
13 biidd 254 . . . . . . . . 9 (𝐵 ∈ V → ((♯‘𝐵) = 1 ↔ (♯‘𝐵) = 1))
14 biidd 254 . . . . . . . . 9 (𝐵 ∈ V → (1 < (♯‘𝐵) ↔ 1 < (♯‘𝐵)))
1512, 13, 143orbi123d 1415 . . . . . . . 8 (𝐵 ∈ V → (((♯‘𝐵) = 0 ∨ (♯‘𝐵) = 1 ∨ 1 < (♯‘𝐵)) ↔ (𝐵 = ∅ ∨ (♯‘𝐵) = 1 ∨ 1 < (♯‘𝐵))))
16 olc 855 . . . . . . . . . . 11 (𝐵 = ∅ → ((♯‘𝐵) = 1 ∨ 𝐵 = ∅))
1716olcd 861 . . . . . . . . . 10 (𝐵 = ∅ → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))
18172a1d 26 . . . . . . . . 9 (𝐵 = ∅ → ((𝐴 = ∅ ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴)) → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))))
19 orc 854 . . . . . . . . . . 11 ((♯‘𝐵) = 1 → ((♯‘𝐵) = 1 ∨ 𝐵 = ∅))
2019olcd 861 . . . . . . . . . 10 ((♯‘𝐵) = 1 → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))
21202a1d 26 . . . . . . . . 9 ((♯‘𝐵) = 1 → ((𝐴 = ∅ ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴)) → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))))
22 olc 855 . . . . . . . . . . . . 13 (𝐴 = ∅ → ((♯‘𝐴) = 1 ∨ 𝐴 = ∅))
2322orcd 860 . . . . . . . . . . . 12 (𝐴 = ∅ → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))
24232a1d 26 . . . . . . . . . . 11 (𝐴 = ∅ → (1 < (♯‘𝐵) → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))))
25 orc 854 . . . . . . . . . . . . 13 ((♯‘𝐴) = 1 → ((♯‘𝐴) = 1 ∨ 𝐴 = ∅))
2625orcd 860 . . . . . . . . . . . 12 ((♯‘𝐴) = 1 → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))
27262a1d 26 . . . . . . . . . . 11 ((♯‘𝐴) = 1 → (1 < (♯‘𝐵) → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))))
28 eqid 2780 . . . . . . . . . . . . . . 15 (Edg‘𝐺) = (Edg‘𝐺)
291, 2, 3, 4, 28frgrwopreglem5 27870 . . . . . . . . . . . . . 14 ((𝐺 ∈ FriendGraph ∧ 1 < (♯‘𝐴) ∧ 1 < (♯‘𝐵)) → ∃𝑎𝐴𝑥𝐴𝑏𝐵𝑦𝐵 ((𝑎𝑥𝑏𝑦) ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑥} ∈ (Edg‘𝐺)) ∧ ({𝑥, 𝑦} ∈ (Edg‘𝐺) ∧ {𝑦, 𝑎} ∈ (Edg‘𝐺))))
30 frgrusgr 27809 . . . . . . . . . . . . . . . 16 (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
31 simplll 763 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐺 ∈ USGraph ∧ (𝑎𝐴𝑥𝐴)) ∧ (𝑏𝐵𝑦𝐵)) ∧ (𝑎𝑥𝑏𝑦)) → 𝐺 ∈ USGraph)
32 elrabi 3592 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑎 ∈ {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾} → 𝑎𝑉)
3332, 3eleq2s 2886 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑎𝐴𝑎𝑉)
3433adantr 473 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑎𝐴𝑥𝐴) → 𝑎𝑉)
3534ad3antlr 719 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐺 ∈ USGraph ∧ (𝑎𝐴𝑥𝐴)) ∧ (𝑏𝐵𝑦𝐵)) ∧ (𝑎𝑥𝑏𝑦)) → 𝑎𝑉)
36 rabidim1 3321 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 ∈ {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾} → 𝑥𝑉)
3736, 3eleq2s 2886 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥𝐴𝑥𝑉)
3837adantl 474 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑎𝐴𝑥𝐴) → 𝑥𝑉)
3938ad3antlr 719 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐺 ∈ USGraph ∧ (𝑎𝐴𝑥𝐴)) ∧ (𝑏𝐵𝑦𝐵)) ∧ (𝑎𝑥𝑏𝑦)) → 𝑥𝑉)
40 simprl 759 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐺 ∈ USGraph ∧ (𝑎𝐴𝑥𝐴)) ∧ (𝑏𝐵𝑦𝐵)) ∧ (𝑎𝑥𝑏𝑦)) → 𝑎𝑥)
41 eldifi 3995 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑏 ∈ (𝑉𝐴) → 𝑏𝑉)
4241, 4eleq2s 2886 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑏𝐵𝑏𝑉)
4342adantr 473 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑏𝐵𝑦𝐵) → 𝑏𝑉)
4443ad2antlr 715 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐺 ∈ USGraph ∧ (𝑎𝐴𝑥𝐴)) ∧ (𝑏𝐵𝑦𝐵)) ∧ (𝑎𝑥𝑏𝑦)) → 𝑏𝑉)
45 eldifi 3995 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ (𝑉𝐴) → 𝑦𝑉)
4645, 4eleq2s 2886 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦𝐵𝑦𝑉)
4746adantl 474 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑏𝐵𝑦𝐵) → 𝑦𝑉)
4847ad2antlr 715 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐺 ∈ USGraph ∧ (𝑎𝐴𝑥𝐴)) ∧ (𝑏𝐵𝑦𝐵)) ∧ (𝑎𝑥𝑏𝑦)) → 𝑦𝑉)
49 simprr 761 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐺 ∈ USGraph ∧ (𝑎𝐴𝑥𝐴)) ∧ (𝑏𝐵𝑦𝐵)) ∧ (𝑎𝑥𝑏𝑦)) → 𝑏𝑦)
501, 284cyclusnfrgr 27841 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐺 ∈ USGraph ∧ (𝑎𝑉𝑥𝑉𝑎𝑥) ∧ (𝑏𝑉𝑦𝑉𝑏𝑦)) → ((({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑥} ∈ (Edg‘𝐺)) ∧ ({𝑥, 𝑦} ∈ (Edg‘𝐺) ∧ {𝑦, 𝑎} ∈ (Edg‘𝐺))) → 𝐺 ∉ FriendGraph ))
5131, 35, 39, 40, 44, 48, 49, 50syl133anc 1374 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐺 ∈ USGraph ∧ (𝑎𝐴𝑥𝐴)) ∧ (𝑏𝐵𝑦𝐵)) ∧ (𝑎𝑥𝑏𝑦)) → ((({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑥} ∈ (Edg‘𝐺)) ∧ ({𝑥, 𝑦} ∈ (Edg‘𝐺) ∧ {𝑦, 𝑎} ∈ (Edg‘𝐺))) → 𝐺 ∉ FriendGraph ))
5251exp4b 423 . . . . . . . . . . . . . . . . . . . . 21 (((𝐺 ∈ USGraph ∧ (𝑎𝐴𝑥𝐴)) ∧ (𝑏𝐵𝑦𝐵)) → ((𝑎𝑥𝑏𝑦) → (({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑥} ∈ (Edg‘𝐺)) → (({𝑥, 𝑦} ∈ (Edg‘𝐺) ∧ {𝑦, 𝑎} ∈ (Edg‘𝐺)) → 𝐺 ∉ FriendGraph ))))
53523impd 1329 . . . . . . . . . . . . . . . . . . . 20 (((𝐺 ∈ USGraph ∧ (𝑎𝐴𝑥𝐴)) ∧ (𝑏𝐵𝑦𝐵)) → (((𝑎𝑥𝑏𝑦) ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑥} ∈ (Edg‘𝐺)) ∧ ({𝑥, 𝑦} ∈ (Edg‘𝐺) ∧ {𝑦, 𝑎} ∈ (Edg‘𝐺))) → 𝐺 ∉ FriendGraph ))
54 df-nel 3076 . . . . . . . . . . . . . . . . . . . . 21 (𝐺 ∉ FriendGraph ↔ ¬ 𝐺 ∈ FriendGraph )
55 pm2.21 121 . . . . . . . . . . . . . . . . . . . . 21 𝐺 ∈ FriendGraph → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅))))
5654, 55sylbi 209 . . . . . . . . . . . . . . . . . . . 20 (𝐺 ∉ FriendGraph → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅))))
5753, 56syl6 35 . . . . . . . . . . . . . . . . . . 19 (((𝐺 ∈ USGraph ∧ (𝑎𝐴𝑥𝐴)) ∧ (𝑏𝐵𝑦𝐵)) → (((𝑎𝑥𝑏𝑦) ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑥} ∈ (Edg‘𝐺)) ∧ ({𝑥, 𝑦} ∈ (Edg‘𝐺) ∧ {𝑦, 𝑎} ∈ (Edg‘𝐺))) → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))))
5857rexlimdvva 3241 . . . . . . . . . . . . . . . . . 18 ((𝐺 ∈ USGraph ∧ (𝑎𝐴𝑥𝐴)) → (∃𝑏𝐵𝑦𝐵 ((𝑎𝑥𝑏𝑦) ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑥} ∈ (Edg‘𝐺)) ∧ ({𝑥, 𝑦} ∈ (Edg‘𝐺) ∧ {𝑦, 𝑎} ∈ (Edg‘𝐺))) → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))))
5958rexlimdvva 3241 . . . . . . . . . . . . . . . . 17 (𝐺 ∈ USGraph → (∃𝑎𝐴𝑥𝐴𝑏𝐵𝑦𝐵 ((𝑎𝑥𝑏𝑦) ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑥} ∈ (Edg‘𝐺)) ∧ ({𝑥, 𝑦} ∈ (Edg‘𝐺) ∧ {𝑦, 𝑎} ∈ (Edg‘𝐺))) → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))))
6059com23 86 . . . . . . . . . . . . . . . 16 (𝐺 ∈ USGraph → (𝐺 ∈ FriendGraph → (∃𝑎𝐴𝑥𝐴𝑏𝐵𝑦𝐵 ((𝑎𝑥𝑏𝑦) ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑥} ∈ (Edg‘𝐺)) ∧ ({𝑥, 𝑦} ∈ (Edg‘𝐺) ∧ {𝑦, 𝑎} ∈ (Edg‘𝐺))) → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))))
6130, 60mpcom 38 . . . . . . . . . . . . . . 15 (𝐺 ∈ FriendGraph → (∃𝑎𝐴𝑥𝐴𝑏𝐵𝑦𝐵 ((𝑎𝑥𝑏𝑦) ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑥} ∈ (Edg‘𝐺)) ∧ ({𝑥, 𝑦} ∈ (Edg‘𝐺) ∧ {𝑦, 𝑎} ∈ (Edg‘𝐺))) → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅))))
62613ad2ant1 1114 . . . . . . . . . . . . . 14 ((𝐺 ∈ FriendGraph ∧ 1 < (♯‘𝐴) ∧ 1 < (♯‘𝐵)) → (∃𝑎𝐴𝑥𝐴𝑏𝐵𝑦𝐵 ((𝑎𝑥𝑏𝑦) ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑥} ∈ (Edg‘𝐺)) ∧ ({𝑥, 𝑦} ∈ (Edg‘𝐺) ∧ {𝑦, 𝑎} ∈ (Edg‘𝐺))) → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅))))
6329, 62mpd 15 . . . . . . . . . . . . 13 ((𝐺 ∈ FriendGraph ∧ 1 < (♯‘𝐴) ∧ 1 < (♯‘𝐵)) → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))
64633exp 1100 . . . . . . . . . . . 12 (𝐺 ∈ FriendGraph → (1 < (♯‘𝐴) → (1 < (♯‘𝐵) → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))))
6564com3l 89 . . . . . . . . . . 11 (1 < (♯‘𝐴) → (1 < (♯‘𝐵) → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))))
6624, 27, 653jaoi 1408 . . . . . . . . . 10 ((𝐴 = ∅ ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴)) → (1 < (♯‘𝐵) → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))))
6766com12 32 . . . . . . . . 9 (1 < (♯‘𝐵) → ((𝐴 = ∅ ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴)) → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))))
6818, 21, 673jaoi 1408 . . . . . . . 8 ((𝐵 = ∅ ∨ (♯‘𝐵) = 1 ∨ 1 < (♯‘𝐵)) → ((𝐴 = ∅ ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴)) → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))))
6915, 68syl6bi 245 . . . . . . 7 (𝐵 ∈ V → (((♯‘𝐵) = 0 ∨ (♯‘𝐵) = 1 ∨ 1 < (♯‘𝐵)) → ((𝐴 = ∅ ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴)) → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅))))))
7011, 69mpd 15 . . . . . 6 (𝐵 ∈ V → ((𝐴 = ∅ ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴)) → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))))
7170com12 32 . . . . 5 ((𝐴 = ∅ ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴)) → (𝐵 ∈ V → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))))
7210, 71syl6bi 245 . . . 4 (𝐴 ∈ V → (((♯‘𝐴) = 0 ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴)) → (𝐵 ∈ V → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅))))))
736, 72mpd 15 . . 3 (𝐴 ∈ V → (𝐵 ∈ V → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))))
7473imp 398 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅))))
755, 74ax-mp 5 1 (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 387  wo 834  w3o 1068  w3a 1069   = wceq 1508  wcel 2051  wne 2969  wnel 3075  wrex 3091  {crab 3094  Vcvv 3417  cdif 3828  c0 4181  {cpr 4446   class class class wbr 4934  cfv 6193  0cc0 10341  1c1 10342   < clt 10480  chash 13511  Vtxcvtx 26499  Edgcedg 26550  USGraphcusgr 26652  VtxDegcvtxdg 26965   FriendGraph cfrgr 27805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2752  ax-rep 5053  ax-sep 5064  ax-nul 5071  ax-pow 5123  ax-pr 5190  ax-un 7285  ax-cnex 10397  ax-resscn 10398  ax-1cn 10399  ax-icn 10400  ax-addcl 10401  ax-addrcl 10402  ax-mulcl 10403  ax-mulrcl 10404  ax-mulcom 10405  ax-addass 10406  ax-mulass 10407  ax-distr 10408  ax-i2m1 10409  ax-1ne0 10410  ax-1rid 10411  ax-rnegex 10412  ax-rrecex 10413  ax-cnre 10414  ax-pre-lttri 10415  ax-pre-lttrn 10416  ax-pre-ltadd 10417  ax-pre-mulgt0 10418
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-fal 1521  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2551  df-eu 2589  df-clab 2761  df-cleq 2773  df-clel 2848  df-nfc 2920  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3419  df-sbc 3684  df-csb 3789  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-pss 3847  df-nul 4182  df-if 4354  df-pw 4427  df-sn 4445  df-pr 4447  df-tp 4449  df-op 4451  df-uni 4718  df-int 4755  df-iun 4799  df-br 4935  df-opab 4997  df-mpt 5014  df-tr 5036  df-id 5316  df-eprel 5321  df-po 5330  df-so 5331  df-fr 5370  df-we 5372  df-xp 5417  df-rel 5418  df-cnv 5419  df-co 5420  df-dm 5421  df-rn 5422  df-res 5423  df-ima 5424  df-pred 5991  df-ord 6037  df-on 6038  df-lim 6039  df-suc 6040  df-iota 6157  df-fun 6195  df-fn 6196  df-f 6197  df-f1 6198  df-fo 6199  df-f1o 6200  df-fv 6201  df-riota 6943  df-ov 6985  df-oprab 6986  df-mpo 6987  df-om 7403  df-1st 7507  df-2nd 7508  df-wrecs 7756  df-recs 7818  df-rdg 7856  df-1o 7911  df-2o 7912  df-oadd 7915  df-er 8095  df-en 8313  df-dom 8314  df-sdom 8315  df-fin 8316  df-dju 9130  df-card 9168  df-pnf 10482  df-mnf 10483  df-xr 10484  df-ltxr 10485  df-le 10486  df-sub 10678  df-neg 10679  df-nn 11446  df-2 11509  df-n0 11714  df-xnn0 11786  df-z 11800  df-uz 12065  df-xadd 12331  df-fz 12715  df-hash 13512  df-edg 26551  df-uhgr 26561  df-ushgr 26562  df-upgr 26585  df-umgr 26586  df-uspgr 26653  df-usgr 26654  df-nbgr 26833  df-vtxdg 26966  df-frgr 27806
This theorem is referenced by:  frgrregorufr0  27873
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