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Theorem frgrwopreg 30342
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 30331 . 2 (𝐴 ∈ V ∧ 𝐵 ∈ V)
6 hashv01gt1 14384 . . . 4 (𝐴 ∈ V → ((♯‘𝐴) = 0 ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴)))
7 hasheq0 14402 . . . . . 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 1437 . . . . 5 (𝐴 ∈ V → (((♯‘𝐴) = 0 ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴)) ↔ (𝐴 = ∅ ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴))))
11 hashv01gt1 14384 . . . . . . 7 (𝐵 ∈ V → ((♯‘𝐵) = 0 ∨ (♯‘𝐵) = 1 ∨ 1 < (♯‘𝐵)))
12 hasheq0 14402 . . . . . . . . 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 1437 . . . . . . . 8 (𝐵 ∈ V → (((♯‘𝐵) = 0 ∨ (♯‘𝐵) = 1 ∨ 1 < (♯‘𝐵)) ↔ (𝐵 = ∅ ∨ (♯‘𝐵) = 1 ∨ 1 < (♯‘𝐵))))
16 olc 869 . . . . . . . . . . 11 (𝐵 = ∅ → ((♯‘𝐵) = 1 ∨ 𝐵 = ∅))
1716olcd 875 . . . . . . . . . 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 868 . . . . . . . . . . 11 ((♯‘𝐵) = 1 → ((♯‘𝐵) = 1 ∨ 𝐵 = ∅))
2019olcd 875 . . . . . . . . . 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 869 . . . . . . . . . . . . 13 (𝐴 = ∅ → ((♯‘𝐴) = 1 ∨ 𝐴 = ∅))
2322orcd 874 . . . . . . . . . . . 12 (𝐴 = ∅ → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))
24232a1d 26 . . . . . . . . . . 11 (𝐴 = ∅ → (1 < (♯‘𝐵) → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))))
25 orc 868 . . . . . . . . . . . . 13 ((♯‘𝐴) = 1 → ((♯‘𝐴) = 1 ∨ 𝐴 = ∅))
2625orcd 874 . . . . . . . . . . . 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 2737 . . . . . . . . . . . . . . 15 (Edg‘𝐺) = (Edg‘𝐺)
291, 2, 3, 4, 28frgrwopreglem5 30340 . . . . . . . . . . . . . 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 30280 . . . . . . . . . . . . . . . 16 (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
31 simplll 775 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐺 ∈ USGraph ∧ (𝑎𝐴𝑥𝐴)) ∧ (𝑏𝐵𝑦𝐵)) ∧ (𝑎𝑥𝑏𝑦)) → 𝐺 ∈ USGraph)
32 elrabi 3687 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑎 ∈ {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾} → 𝑎𝑉)
3332, 3eleq2s 2859 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑎𝐴𝑎𝑉)
3433adantr 480 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑎𝐴𝑥𝐴) → 𝑎𝑉)
3534ad3antlr 731 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐺 ∈ USGraph ∧ (𝑎𝐴𝑥𝐴)) ∧ (𝑏𝐵𝑦𝐵)) ∧ (𝑎𝑥𝑏𝑦)) → 𝑎𝑉)
36 rabidim1 3459 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 ∈ {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾} → 𝑥𝑉)
3736, 3eleq2s 2859 . . . . . . . . . . . . . . . . . . . . . . . . 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 4131 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑏 ∈ (𝑉𝐴) → 𝑏𝑉)
4241, 4eleq2s 2859 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑏𝐵𝑏𝑉)
4342adantr 480 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑏𝐵𝑦𝐵) → 𝑏𝑉)
4443ad2antlr 727 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐺 ∈ USGraph ∧ (𝑎𝐴𝑥𝐴)) ∧ (𝑏𝐵𝑦𝐵)) ∧ (𝑎𝑥𝑏𝑦)) → 𝑏𝑉)
45 eldifi 4131 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ (𝑉𝐴) → 𝑦𝑉)
4645, 4eleq2s 2859 . . . . . . . . . . . . . . . . . . . . . . . . 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 30311 . . . . . . . . . . . . . . . . . . . . . . 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 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 1349 . . . . . . . . . . . . . . . . . . . 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 3047 . . . . . . . . . . . . . . . . . . . . 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 3213 . . . . . . . . . . . . . . . . . 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 3213 . . . . . . . . . . . . . . . . 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 1134 . . . . . . . . . . . . . 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 1120 . . . . . . . . . . . 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 1430 . . . . . . . . . 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 1430 . . . . . . . 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 848  w3o 1086  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wnel 3046  wrex 3070  {crab 3436  Vcvv 3480  cdif 3948  c0 4333  {cpr 4628   class class class wbr 5143  cfv 6561  0cc0 11155  1c1 11156   < clt 11295  chash 14369  Vtxcvtx 29013  Edgcedg 29064  USGraphcusgr 29166  VtxDegcvtxdg 29483   FriendGraph cfrgr 30277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-n0 12527  df-xnn0 12600  df-z 12614  df-uz 12879  df-xadd 13155  df-fz 13548  df-hash 14370  df-edg 29065  df-uhgr 29075  df-ushgr 29076  df-upgr 29099  df-umgr 29100  df-uspgr 29167  df-usgr 29168  df-nbgr 29350  df-vtxdg 29484  df-frgr 30278
This theorem is referenced by:  frgrregorufr0  30343
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