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Theorem frgrwopreg 28419
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 28408 . 2 (𝐴 ∈ V ∧ 𝐵 ∈ V)
6 hashv01gt1 13924 . . . 4 (𝐴 ∈ V → ((♯‘𝐴) = 0 ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴)))
7 hasheq0 13943 . . . . . 6 (𝐴 ∈ V → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅))
8 biidd 265 . . . . . 6 (𝐴 ∈ V → ((♯‘𝐴) = 1 ↔ (♯‘𝐴) = 1))
9 biidd 265 . . . . . 6 (𝐴 ∈ V → (1 < (♯‘𝐴) ↔ 1 < (♯‘𝐴)))
107, 8, 93orbi123d 1437 . . . . 5 (𝐴 ∈ V → (((♯‘𝐴) = 0 ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴)) ↔ (𝐴 = ∅ ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴))))
11 hashv01gt1 13924 . . . . . . 7 (𝐵 ∈ V → ((♯‘𝐵) = 0 ∨ (♯‘𝐵) = 1 ∨ 1 < (♯‘𝐵)))
12 hasheq0 13943 . . . . . . . . 9 (𝐵 ∈ V → ((♯‘𝐵) = 0 ↔ 𝐵 = ∅))
13 biidd 265 . . . . . . . . 9 (𝐵 ∈ V → ((♯‘𝐵) = 1 ↔ (♯‘𝐵) = 1))
14 biidd 265 . . . . . . . . 9 (𝐵 ∈ V → (1 < (♯‘𝐵) ↔ 1 < (♯‘𝐵)))
1512, 13, 143orbi123d 1437 . . . . . . . 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 2738 . . . . . . . . . . . . . . 15 (Edg‘𝐺) = (Edg‘𝐺)
291, 2, 3, 4, 28frgrwopreglem5 28417 . . . . . . . . . . . . . 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 28357 . . . . . . . . . . . . . . . 16 (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
31 simplll 775 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐺 ∈ USGraph ∧ (𝑎𝐴𝑥𝐴)) ∧ (𝑏𝐵𝑦𝐵)) ∧ (𝑎𝑥𝑏𝑦)) → 𝐺 ∈ USGraph)
32 elrabi 3603 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑎 ∈ {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾} → 𝑎𝑉)
3332, 3eleq2s 2857 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑎𝐴𝑎𝑉)
3433adantr 484 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑎𝐴𝑥𝐴) → 𝑎𝑉)
3534ad3antlr 731 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐺 ∈ USGraph ∧ (𝑎𝐴𝑥𝐴)) ∧ (𝑏𝐵𝑦𝐵)) ∧ (𝑎𝑥𝑏𝑦)) → 𝑎𝑉)
36 rabidim1 3299 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 ∈ {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾} → 𝑥𝑉)
3736, 3eleq2s 2857 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥𝐴𝑥𝑉)
3837adantl 485 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑎𝐴𝑥𝐴) → 𝑥𝑉)
3938ad3antlr 731 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐺 ∈ USGraph ∧ (𝑎𝐴𝑥𝐴)) ∧ (𝑏𝐵𝑦𝐵)) ∧ (𝑎𝑥𝑏𝑦)) → 𝑥𝑉)
40 simprl 771 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐺 ∈ USGraph ∧ (𝑎𝐴𝑥𝐴)) ∧ (𝑏𝐵𝑦𝐵)) ∧ (𝑎𝑥𝑏𝑦)) → 𝑎𝑥)
41 eldifi 4050 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑏 ∈ (𝑉𝐴) → 𝑏𝑉)
4241, 4eleq2s 2857 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑏𝐵𝑏𝑉)
4342adantr 484 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑏𝐵𝑦𝐵) → 𝑏𝑉)
4443ad2antlr 727 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐺 ∈ USGraph ∧ (𝑎𝐴𝑥𝐴)) ∧ (𝑏𝐵𝑦𝐵)) ∧ (𝑎𝑥𝑏𝑦)) → 𝑏𝑉)
45 eldifi 4050 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ (𝑉𝐴) → 𝑦𝑉)
4645, 4eleq2s 2857 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦𝐵𝑦𝑉)
4746adantl 485 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑏𝐵𝑦𝐵) → 𝑦𝑉)
4847ad2antlr 727 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐺 ∈ USGraph ∧ (𝑎𝐴𝑥𝐴)) ∧ (𝑏𝐵𝑦𝐵)) ∧ (𝑎𝑥𝑏𝑦)) → 𝑦𝑉)
49 simprr 773 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐺 ∈ USGraph ∧ (𝑎𝐴𝑥𝐴)) ∧ (𝑏𝐵𝑦𝐵)) ∧ (𝑎𝑥𝑏𝑦)) → 𝑏𝑦)
501, 284cyclusnfrgr 28388 . . . . . . . . . . . . . . . . . . . . . . 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 1395 . . . . . . . . . . . . . . . . . . . . . 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 434 . . . . . . . . . . . . . . . . . . . . 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 1350 . . . . . . . . . . . . . . . . . . . 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 3048 . . . . . . . . . . . . . . . . . . . . 21 (𝐺 ∉ FriendGraph ↔ ¬ 𝐺 ∈ FriendGraph )
55 pm2.21 123 . . . . . . . . . . . . . . . . . . . . 21 𝐺 ∈ FriendGraph → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅))))
5654, 55sylbi 220 . . . . . . . . . . . . . . . . . . . 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 3220 . . . . . . . . . . . . . . . . . 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 3220 . . . . . . . . . . . . . . . . 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 1135 . . . . . . . . . . . . . 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 1121 . . . . . . . . . . . 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 1429 . . . . . . . . . 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 1429 . . . . . . . 8 ((𝐵 = ∅ ∨ (♯‘𝐵) = 1 ∨ 1 < (♯‘𝐵)) → ((𝐴 = ∅ ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴)) → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))))
6915, 68syl6bi 256 . . . . . . 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 256 . . . 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 410 . 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 399  wo 847  w3o 1088  w3a 1089   = wceq 1543  wcel 2111  wne 2941  wnel 3047  wrex 3063  {crab 3066  Vcvv 3415  cdif 3872  c0 4246  {cpr 4552   class class class wbr 5062  cfv 6389  0cc0 10742  1c1 10743   < clt 10880  chash 13909  Vtxcvtx 27100  Edgcedg 27151  USGraphcusgr 27253  VtxDegcvtxdg 27566   FriendGraph cfrgr 28354
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-rep 5188  ax-sep 5201  ax-nul 5208  ax-pow 5267  ax-pr 5331  ax-un 7532  ax-cnex 10798  ax-resscn 10799  ax-1cn 10800  ax-icn 10801  ax-addcl 10802  ax-addrcl 10803  ax-mulcl 10804  ax-mulrcl 10805  ax-mulcom 10806  ax-addass 10807  ax-mulass 10808  ax-distr 10809  ax-i2m1 10810  ax-1ne0 10811  ax-1rid 10812  ax-rnegex 10813  ax-rrecex 10814  ax-cnre 10815  ax-pre-lttri 10816  ax-pre-lttrn 10817  ax-pre-ltadd 10818  ax-pre-mulgt0 10819
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3067  df-rex 3068  df-reu 3069  df-rmo 3070  df-rab 3071  df-v 3417  df-sbc 3704  df-csb 3821  df-dif 3878  df-un 3880  df-in 3882  df-ss 3892  df-pss 3894  df-nul 4247  df-if 4449  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4829  df-int 4869  df-iun 4915  df-br 5063  df-opab 5125  df-mpt 5145  df-tr 5171  df-id 5464  df-eprel 5469  df-po 5477  df-so 5478  df-fr 5518  df-we 5520  df-xp 5566  df-rel 5567  df-cnv 5568  df-co 5569  df-dm 5570  df-rn 5571  df-res 5572  df-ima 5573  df-pred 6169  df-ord 6225  df-on 6226  df-lim 6227  df-suc 6228  df-iota 6347  df-fun 6391  df-fn 6392  df-f 6393  df-f1 6394  df-fo 6395  df-f1o 6396  df-fv 6397  df-riota 7179  df-ov 7225  df-oprab 7226  df-mpo 7227  df-om 7654  df-1st 7770  df-2nd 7771  df-wrecs 8056  df-recs 8117  df-rdg 8155  df-1o 8211  df-2o 8212  df-oadd 8215  df-er 8400  df-en 8636  df-dom 8637  df-sdom 8638  df-fin 8639  df-dju 9530  df-card 9568  df-pnf 10882  df-mnf 10883  df-xr 10884  df-ltxr 10885  df-le 10886  df-sub 11077  df-neg 11078  df-nn 11844  df-2 11906  df-n0 12104  df-xnn0 12176  df-z 12190  df-uz 12452  df-xadd 12718  df-fz 13109  df-hash 13910  df-edg 27152  df-uhgr 27162  df-ushgr 27163  df-upgr 27186  df-umgr 27187  df-uspgr 27254  df-usgr 27255  df-nbgr 27434  df-vtxdg 27567  df-frgr 28355
This theorem is referenced by:  frgrregorufr0  28420
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