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Theorem frgrwopreg 27398
 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 27387 . 2 (𝐴 ∈ V ∧ 𝐵 ∈ V)
6 hashv01gt1 13248 . . . 4 (𝐴 ∈ V → ((♯‘𝐴) = 0 ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴)))
7 hasheq0 13267 . . . . . 6 (𝐴 ∈ V → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅))
8 biidd 252 . . . . . 6 (𝐴 ∈ V → ((♯‘𝐴) = 1 ↔ (♯‘𝐴) = 1))
9 biidd 252 . . . . . 6 (𝐴 ∈ V → (1 < (♯‘𝐴) ↔ 1 < (♯‘𝐴)))
107, 8, 93orbi123d 1511 . . . . 5 (𝐴 ∈ V → (((♯‘𝐴) = 0 ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴)) ↔ (𝐴 = ∅ ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴))))
11 hashv01gt1 13248 . . . . . . 7 (𝐵 ∈ V → ((♯‘𝐵) = 0 ∨ (♯‘𝐵) = 1 ∨ 1 < (♯‘𝐵)))
12 hasheq0 13267 . . . . . . . . 9 (𝐵 ∈ V → ((♯‘𝐵) = 0 ↔ 𝐵 = ∅))
13 biidd 252 . . . . . . . . 9 (𝐵 ∈ V → ((♯‘𝐵) = 1 ↔ (♯‘𝐵) = 1))
14 biidd 252 . . . . . . . . 9 (𝐵 ∈ V → (1 < (♯‘𝐵) ↔ 1 < (♯‘𝐵)))
1512, 13, 143orbi123d 1511 . . . . . . . 8 (𝐵 ∈ V → (((♯‘𝐵) = 0 ∨ (♯‘𝐵) = 1 ∨ 1 < (♯‘𝐵)) ↔ (𝐵 = ∅ ∨ (♯‘𝐵) = 1 ∨ 1 < (♯‘𝐵))))
16 olc 398 . . . . . . . . . . 11 (𝐵 = ∅ → ((♯‘𝐵) = 1 ∨ 𝐵 = ∅))
1716olcd 407 . . . . . . . . . 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 399 . . . . . . . . . . 11 ((♯‘𝐵) = 1 → ((♯‘𝐵) = 1 ∨ 𝐵 = ∅))
2019olcd 407 . . . . . . . . . 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 398 . . . . . . . . . . . . 13 (𝐴 = ∅ → ((♯‘𝐴) = 1 ∨ 𝐴 = ∅))
2322orcd 406 . . . . . . . . . . . 12 (𝐴 = ∅ → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))
24232a1d 26 . . . . . . . . . . 11 (𝐴 = ∅ → (1 < (♯‘𝐵) → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))))
25 orc 399 . . . . . . . . . . . . 13 ((♯‘𝐴) = 1 → ((♯‘𝐴) = 1 ∨ 𝐴 = ∅))
2625orcd 406 . . . . . . . . . . . 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 2724 . . . . . . . . . . . . . . 15 (Edg‘𝐺) = (Edg‘𝐺)
291, 2, 3, 4, 28frgrwopreglem5 27396 . . . . . . . . . . . . . 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 27335 . . . . . . . . . . . . . . . 16 (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
31 simplll 815 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐺 ∈ USGraph ∧ (𝑎𝐴𝑥𝐴)) ∧ (𝑏𝐵𝑦𝐵)) ∧ (𝑎𝑥𝑏𝑦)) → 𝐺 ∈ USGraph)
32 elrabi 3464 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑎 ∈ {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾} → 𝑎𝑉)
3332, 3eleq2s 2821 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑎𝐴𝑎𝑉)
3433adantr 472 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑎𝐴𝑥𝐴) → 𝑎𝑉)
3534ad3antlr 769 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐺 ∈ USGraph ∧ (𝑎𝐴𝑥𝐴)) ∧ (𝑏𝐵𝑦𝐵)) ∧ (𝑎𝑥𝑏𝑦)) → 𝑎𝑉)
36 rabidim1 3220 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 ∈ {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾} → 𝑥𝑉)
3736, 3eleq2s 2821 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥𝐴𝑥𝑉)
3837adantl 473 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑎𝐴𝑥𝐴) → 𝑥𝑉)
3938ad3antlr 769 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐺 ∈ USGraph ∧ (𝑎𝐴𝑥𝐴)) ∧ (𝑏𝐵𝑦𝐵)) ∧ (𝑎𝑥𝑏𝑦)) → 𝑥𝑉)
40 simprl 811 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐺 ∈ USGraph ∧ (𝑎𝐴𝑥𝐴)) ∧ (𝑏𝐵𝑦𝐵)) ∧ (𝑎𝑥𝑏𝑦)) → 𝑎𝑥)
41 eldifi 3840 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑏 ∈ (𝑉𝐴) → 𝑏𝑉)
4241, 4eleq2s 2821 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑏𝐵𝑏𝑉)
4342adantr 472 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑏𝐵𝑦𝐵) → 𝑏𝑉)
4443ad2antlr 765 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐺 ∈ USGraph ∧ (𝑎𝐴𝑥𝐴)) ∧ (𝑏𝐵𝑦𝐵)) ∧ (𝑎𝑥𝑏𝑦)) → 𝑏𝑉)
45 eldifi 3840 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ (𝑉𝐴) → 𝑦𝑉)
4645, 4eleq2s 2821 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦𝐵𝑦𝑉)
4746adantl 473 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑏𝐵𝑦𝐵) → 𝑦𝑉)
4847ad2antlr 765 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐺 ∈ USGraph ∧ (𝑎𝐴𝑥𝐴)) ∧ (𝑏𝐵𝑦𝐵)) ∧ (𝑎𝑥𝑏𝑦)) → 𝑦𝑉)
49 simprr 813 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐺 ∈ USGraph ∧ (𝑎𝐴𝑥𝐴)) ∧ (𝑏𝐵𝑦𝐵)) ∧ (𝑎𝑥𝑏𝑦)) → 𝑏𝑦)
501, 284cyclusnfrgr 27367 . . . . . . . . . . . . . . . . . . . . . . 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 1462 . . . . . . . . . . . . . . . . . . . . . 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 633 . . . . . . . . . . . . . . . . . . . . 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 1405 . . . . . . . . . . . . . . . . . . . 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 3000 . . . . . . . . . . . . . . . . . . . . 21 (𝐺 ∉ FriendGraph ↔ ¬ 𝐺 ∈ FriendGraph )
55 pm2.21 120 . . . . . . . . . . . . . . . . . . . . 21 𝐺 ∈ FriendGraph → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅))))
5654, 55sylbi 207 . . . . . . . . . . . . . . . . . . . 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 3140 . . . . . . . . . . . . . . . . . 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 3140 . . . . . . . . . . . . . . . . 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 1125 . . . . . . . . . . . . . 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 1112 . . . . . . . . . . . 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 1504 . . . . . . . . . 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 1504 . . . . . . . 8 ((𝐵 = ∅ ∨ (♯‘𝐵) = 1 ∨ 1 < (♯‘𝐵)) → ((𝐴 = ∅ ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴)) → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))))
6915, 68syl6bi 243 . . . . . . 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 243 . . . 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 444 . 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   ∨ wo 382   ∧ wa 383   ∨ w3o 1071   ∧ w3a 1072   = wceq 1596   ∈ wcel 2103   ≠ wne 2896   ∉ wnel 2999  ∃wrex 3015  {crab 3018  Vcvv 3304   ∖ cdif 3677  ∅c0 4023  {cpr 4287   class class class wbr 4760  ‘cfv 6001  0cc0 10049  1c1 10050   < clt 10187  ♯chash 13232  Vtxcvtx 25994  Edgcedg 26059  USGraphcusgr 26164  VtxDegcvtxdg 26492   FriendGraph cfrgr 27331 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066  ax-cnex 10105  ax-resscn 10106  ax-1cn 10107  ax-icn 10108  ax-addcl 10109  ax-addrcl 10110  ax-mulcl 10111  ax-mulrcl 10112  ax-mulcom 10113  ax-addass 10114  ax-mulass 10115  ax-distr 10116  ax-i2m1 10117  ax-1ne0 10118  ax-1rid 10119  ax-rnegex 10120  ax-rrecex 10121  ax-cnre 10122  ax-pre-lttri 10123  ax-pre-lttrn 10124  ax-pre-ltadd 10125  ax-pre-mulgt0 10126 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-fal 1602  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-nel 3000  df-ral 3019  df-rex 3020  df-reu 3021  df-rmo 3022  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-int 4584  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-pred 5793  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-riota 6726  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-om 7183  df-1st 7285  df-2nd 7286  df-wrecs 7527  df-recs 7588  df-rdg 7626  df-1o 7680  df-2o 7681  df-oadd 7684  df-er 7862  df-en 8073  df-dom 8074  df-sdom 8075  df-fin 8076  df-card 8878  df-cda 9103  df-pnf 10189  df-mnf 10190  df-xr 10191  df-ltxr 10192  df-le 10193  df-sub 10381  df-neg 10382  df-nn 11134  df-2 11192  df-n0 11406  df-xnn0 11477  df-z 11491  df-uz 11801  df-xadd 12061  df-fz 12441  df-hash 13233  df-edg 26060  df-uhgr 26073  df-ushgr 26074  df-upgr 26097  df-umgr 26098  df-uspgr 26165  df-usgr 26166  df-nbgr 26345  df-vtxdg 26493  df-frgr 27332 This theorem is referenced by:  frgrregorufr0  27399
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