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Theorem frgrwopreg 30352
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 30341 . 2 (𝐴 ∈ V ∧ 𝐵 ∈ V)
6 hashv01gt1 14381 . . . 4 (𝐴 ∈ V → ((♯‘𝐴) = 0 ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴)))
7 hasheq0 14399 . . . . . 6 (𝐴 ∈ V → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅))
8 biidd 262 . . . . . 6 (𝐴 ∈ V → ((♯‘𝐴) = 1 ↔ (♯‘𝐴) = 1))
9 biidd 262 . . . . . 6 (𝐴 ∈ V → (1 < (♯‘𝐴) ↔ 1 < (♯‘𝐴)))
107, 8, 93orbi123d 1434 . . . . 5 (𝐴 ∈ V → (((♯‘𝐴) = 0 ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴)) ↔ (𝐴 = ∅ ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴))))
11 hashv01gt1 14381 . . . . . . 7 (𝐵 ∈ V → ((♯‘𝐵) = 0 ∨ (♯‘𝐵) = 1 ∨ 1 < (♯‘𝐵)))
12 hasheq0 14399 . . . . . . . . 9 (𝐵 ∈ V → ((♯‘𝐵) = 0 ↔ 𝐵 = ∅))
13 biidd 262 . . . . . . . . 9 (𝐵 ∈ V → ((♯‘𝐵) = 1 ↔ (♯‘𝐵) = 1))
14 biidd 262 . . . . . . . . 9 (𝐵 ∈ V → (1 < (♯‘𝐵) ↔ 1 < (♯‘𝐵)))
1512, 13, 143orbi123d 1434 . . . . . . . 8 (𝐵 ∈ V → (((♯‘𝐵) = 0 ∨ (♯‘𝐵) = 1 ∨ 1 < (♯‘𝐵)) ↔ (𝐵 = ∅ ∨ (♯‘𝐵) = 1 ∨ 1 < (♯‘𝐵))))
16 olc 868 . . . . . . . . . . 11 (𝐵 = ∅ → ((♯‘𝐵) = 1 ∨ 𝐵 = ∅))
1716olcd 874 . . . . . . . . . 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 867 . . . . . . . . . . 11 ((♯‘𝐵) = 1 → ((♯‘𝐵) = 1 ∨ 𝐵 = ∅))
2019olcd 874 . . . . . . . . . 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 868 . . . . . . . . . . . . 13 (𝐴 = ∅ → ((♯‘𝐴) = 1 ∨ 𝐴 = ∅))
2322orcd 873 . . . . . . . . . . . 12 (𝐴 = ∅ → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))
24232a1d 26 . . . . . . . . . . 11 (𝐴 = ∅ → (1 < (♯‘𝐵) → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))))
25 orc 867 . . . . . . . . . . . . 13 ((♯‘𝐴) = 1 → ((♯‘𝐴) = 1 ∨ 𝐴 = ∅))
2625orcd 873 . . . . . . . . . . . 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 2735 . . . . . . . . . . . . . . 15 (Edg‘𝐺) = (Edg‘𝐺)
291, 2, 3, 4, 28frgrwopreglem5 30350 . . . . . . . . . . . . . 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 30290 . . . . . . . . . . . . . . . 16 (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
31 simplll 775 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐺 ∈ USGraph ∧ (𝑎𝐴𝑥𝐴)) ∧ (𝑏𝐵𝑦𝐵)) ∧ (𝑎𝑥𝑏𝑦)) → 𝐺 ∈ USGraph)
32 elrabi 3690 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑎 ∈ {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾} → 𝑎𝑉)
3332, 3eleq2s 2857 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑎𝐴𝑎𝑉)
3433adantr 480 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑎𝐴𝑥𝐴) → 𝑎𝑉)
3534ad3antlr 731 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐺 ∈ USGraph ∧ (𝑎𝐴𝑥𝐴)) ∧ (𝑏𝐵𝑦𝐵)) ∧ (𝑎𝑥𝑏𝑦)) → 𝑎𝑉)
36 rabidim1 3456 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 ∈ {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾} → 𝑥𝑉)
3736, 3eleq2s 2857 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥𝐴𝑥𝑉)
3837adantl 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑎𝐴𝑥𝐴) → 𝑥𝑉)
3938ad3antlr 731 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐺 ∈ USGraph ∧ (𝑎𝐴𝑥𝐴)) ∧ (𝑏𝐵𝑦𝐵)) ∧ (𝑎𝑥𝑏𝑦)) → 𝑥𝑉)
40 simprl 771 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐺 ∈ USGraph ∧ (𝑎𝐴𝑥𝐴)) ∧ (𝑏𝐵𝑦𝐵)) ∧ (𝑎𝑥𝑏𝑦)) → 𝑎𝑥)
41 eldifi 4141 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑏 ∈ (𝑉𝐴) → 𝑏𝑉)
4241, 4eleq2s 2857 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑏𝐵𝑏𝑉)
4342adantr 480 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑏𝐵𝑦𝐵) → 𝑏𝑉)
4443ad2antlr 727 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐺 ∈ USGraph ∧ (𝑎𝐴𝑥𝐴)) ∧ (𝑏𝐵𝑦𝐵)) ∧ (𝑎𝑥𝑏𝑦)) → 𝑏𝑉)
45 eldifi 4141 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ (𝑉𝐴) → 𝑦𝑉)
4645, 4eleq2s 2857 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦𝐵𝑦𝑉)
4746adantl 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑏𝐵𝑦𝐵) → 𝑦𝑉)
4847ad2antlr 727 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐺 ∈ USGraph ∧ (𝑎𝐴𝑥𝐴)) ∧ (𝑏𝐵𝑦𝐵)) ∧ (𝑎𝑥𝑏𝑦)) → 𝑦𝑉)
49 simprr 773 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐺 ∈ USGraph ∧ (𝑎𝐴𝑥𝐴)) ∧ (𝑏𝐵𝑦𝐵)) ∧ (𝑎𝑥𝑏𝑦)) → 𝑏𝑦)
501, 284cyclusnfrgr 30321 . . . . . . . . . . . . . . . . . . . . . . 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 1392 . . . . . . . . . . . . . . . . . . . . . 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 430 . . . . . . . . . . . . . . . . . . . . 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 1347 . . . . . . . . . . . . . . . . . . . 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 3045 . . . . . . . . . . . . . . . . . . . . 21 (𝐺 ∉ FriendGraph ↔ ¬ 𝐺 ∈ FriendGraph )
55 pm2.21 123 . . . . . . . . . . . . . . . . . . . . 21 𝐺 ∈ FriendGraph → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅))))
5654, 55sylbi 217 . . . . . . . . . . . . . . . . . . . 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 3211 . . . . . . . . . . . . . . . . . 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 3211 . . . . . . . . . . . . . . . . 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 1132 . . . . . . . . . . . . . 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 1118 . . . . . . . . . . . 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 1427 . . . . . . . . . 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 1427 . . . . . . . 8 ((𝐵 = ∅ ∨ (♯‘𝐵) = 1 ∨ 1 < (♯‘𝐵)) → ((𝐴 = ∅ ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴)) → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))))
6915, 68biimtrdi 253 . . . . . . 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, 71biimtrdi 253 . . . 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 406 . 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 395  wo 847  w3o 1085  w3a 1086   = wceq 1537  wcel 2106  wne 2938  wnel 3044  wrex 3068  {crab 3433  Vcvv 3478  cdif 3960  c0 4339  {cpr 4633   class class class wbr 5148  cfv 6563  0cc0 11153  1c1 11154   < clt 11293  chash 14366  Vtxcvtx 29028  Edgcedg 29079  USGraphcusgr 29181  VtxDegcvtxdg 29498   FriendGraph cfrgr 30287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-oadd 8509  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-dju 9939  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-n0 12525  df-xnn0 12598  df-z 12612  df-uz 12877  df-xadd 13153  df-fz 13545  df-hash 14367  df-edg 29080  df-uhgr 29090  df-ushgr 29091  df-upgr 29114  df-umgr 29115  df-uspgr 29182  df-usgr 29183  df-nbgr 29365  df-vtxdg 29499  df-frgr 30288
This theorem is referenced by:  frgrregorufr0  30353
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