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Theorem frgrwopreg 29309
Description: In a friendship graph there are either no vertices (𝐴 = βˆ…) or exactly one vertex ((β™―β€˜π΄) = 1) having degree 𝐾, or all (𝐡 = βˆ…) or all except one vertices ((β™―β€˜π΅) = 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 = (Vtxβ€˜πΊ)
frgrwopreg.d 𝐷 = (VtxDegβ€˜πΊ)
frgrwopreg.a 𝐴 = {π‘₯ ∈ 𝑉 ∣ (π·β€˜π‘₯) = 𝐾}
frgrwopreg.b 𝐡 = (𝑉 βˆ– 𝐴)
Assertion
Ref Expression
frgrwopreg (𝐺 ∈ FriendGraph β†’ (((β™―β€˜π΄) = 1 ∨ 𝐴 = βˆ…) ∨ ((β™―β€˜π΅) = 1 ∨ 𝐡 = βˆ…)))
Distinct variable groups:   π‘₯,𝑉   π‘₯,𝐴   π‘₯,𝐺   π‘₯,𝐾   π‘₯,𝐷   π‘₯,𝐡

Proof of Theorem frgrwopreg
Dummy variables π‘Ž 𝑏 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frgrwopreg.v . . 3 𝑉 = (Vtxβ€˜πΊ)
2 frgrwopreg.d . . 3 𝐷 = (VtxDegβ€˜πΊ)
3 frgrwopreg.a . . 3 𝐴 = {π‘₯ ∈ 𝑉 ∣ (π·β€˜π‘₯) = 𝐾}
4 frgrwopreg.b . . 3 𝐡 = (𝑉 βˆ– 𝐴)
51, 2, 3, 4frgrwopreglem1 29298 . 2 (𝐴 ∈ V ∧ 𝐡 ∈ V)
6 hashv01gt1 14252 . . . 4 (𝐴 ∈ V β†’ ((β™―β€˜π΄) = 0 ∨ (β™―β€˜π΄) = 1 ∨ 1 < (β™―β€˜π΄)))
7 hasheq0 14270 . . . . . 6 (𝐴 ∈ V β†’ ((β™―β€˜π΄) = 0 ↔ 𝐴 = βˆ…))
8 biidd 262 . . . . . 6 (𝐴 ∈ V β†’ ((β™―β€˜π΄) = 1 ↔ (β™―β€˜π΄) = 1))
9 biidd 262 . . . . . 6 (𝐴 ∈ V β†’ (1 < (β™―β€˜π΄) ↔ 1 < (β™―β€˜π΄)))
107, 8, 93orbi123d 1436 . . . . 5 (𝐴 ∈ V β†’ (((β™―β€˜π΄) = 0 ∨ (β™―β€˜π΄) = 1 ∨ 1 < (β™―β€˜π΄)) ↔ (𝐴 = βˆ… ∨ (β™―β€˜π΄) = 1 ∨ 1 < (β™―β€˜π΄))))
11 hashv01gt1 14252 . . . . . . 7 (𝐡 ∈ V β†’ ((β™―β€˜π΅) = 0 ∨ (β™―β€˜π΅) = 1 ∨ 1 < (β™―β€˜π΅)))
12 hasheq0 14270 . . . . . . . . 9 (𝐡 ∈ V β†’ ((β™―β€˜π΅) = 0 ↔ 𝐡 = βˆ…))
13 biidd 262 . . . . . . . . 9 (𝐡 ∈ V β†’ ((β™―β€˜π΅) = 1 ↔ (β™―β€˜π΅) = 1))
14 biidd 262 . . . . . . . . 9 (𝐡 ∈ V β†’ (1 < (β™―β€˜π΅) ↔ 1 < (β™―β€˜π΅)))
1512, 13, 143orbi123d 1436 . . . . . . . 8 (𝐡 ∈ V β†’ (((β™―β€˜π΅) = 0 ∨ (β™―β€˜π΅) = 1 ∨ 1 < (β™―β€˜π΅)) ↔ (𝐡 = βˆ… ∨ (β™―β€˜π΅) = 1 ∨ 1 < (β™―β€˜π΅))))
16 olc 867 . . . . . . . . . . 11 (𝐡 = βˆ… β†’ ((β™―β€˜π΅) = 1 ∨ 𝐡 = βˆ…))
1716olcd 873 . . . . . . . . . 10 (𝐡 = βˆ… β†’ (((β™―β€˜π΄) = 1 ∨ 𝐴 = βˆ…) ∨ ((β™―β€˜π΅) = 1 ∨ 𝐡 = βˆ…)))
18172a1d 26 . . . . . . . . 9 (𝐡 = βˆ… β†’ ((𝐴 = βˆ… ∨ (β™―β€˜π΄) = 1 ∨ 1 < (β™―β€˜π΄)) β†’ (𝐺 ∈ FriendGraph β†’ (((β™―β€˜π΄) = 1 ∨ 𝐴 = βˆ…) ∨ ((β™―β€˜π΅) = 1 ∨ 𝐡 = βˆ…)))))
19 orc 866 . . . . . . . . . . 11 ((β™―β€˜π΅) = 1 β†’ ((β™―β€˜π΅) = 1 ∨ 𝐡 = βˆ…))
2019olcd 873 . . . . . . . . . 10 ((β™―β€˜π΅) = 1 β†’ (((β™―β€˜π΄) = 1 ∨ 𝐴 = βˆ…) ∨ ((β™―β€˜π΅) = 1 ∨ 𝐡 = βˆ…)))
21202a1d 26 . . . . . . . . 9 ((β™―β€˜π΅) = 1 β†’ ((𝐴 = βˆ… ∨ (β™―β€˜π΄) = 1 ∨ 1 < (β™―β€˜π΄)) β†’ (𝐺 ∈ FriendGraph β†’ (((β™―β€˜π΄) = 1 ∨ 𝐴 = βˆ…) ∨ ((β™―β€˜π΅) = 1 ∨ 𝐡 = βˆ…)))))
22 olc 867 . . . . . . . . . . . . 13 (𝐴 = βˆ… β†’ ((β™―β€˜π΄) = 1 ∨ 𝐴 = βˆ…))
2322orcd 872 . . . . . . . . . . . 12 (𝐴 = βˆ… β†’ (((β™―β€˜π΄) = 1 ∨ 𝐴 = βˆ…) ∨ ((β™―β€˜π΅) = 1 ∨ 𝐡 = βˆ…)))
24232a1d 26 . . . . . . . . . . 11 (𝐴 = βˆ… β†’ (1 < (β™―β€˜π΅) β†’ (𝐺 ∈ FriendGraph β†’ (((β™―β€˜π΄) = 1 ∨ 𝐴 = βˆ…) ∨ ((β™―β€˜π΅) = 1 ∨ 𝐡 = βˆ…)))))
25 orc 866 . . . . . . . . . . . . 13 ((β™―β€˜π΄) = 1 β†’ ((β™―β€˜π΄) = 1 ∨ 𝐴 = βˆ…))
2625orcd 872 . . . . . . . . . . . 12 ((β™―β€˜π΄) = 1 β†’ (((β™―β€˜π΄) = 1 ∨ 𝐴 = βˆ…) ∨ ((β™―β€˜π΅) = 1 ∨ 𝐡 = βˆ…)))
27262a1d 26 . . . . . . . . . . 11 ((β™―β€˜π΄) = 1 β†’ (1 < (β™―β€˜π΅) β†’ (𝐺 ∈ FriendGraph β†’ (((β™―β€˜π΄) = 1 ∨ 𝐴 = βˆ…) ∨ ((β™―β€˜π΅) = 1 ∨ 𝐡 = βˆ…)))))
28 eqid 2737 . . . . . . . . . . . . . . 15 (Edgβ€˜πΊ) = (Edgβ€˜πΊ)
291, 2, 3, 4, 28frgrwopreglem5 29307 . . . . . . . . . . . . . 14 ((𝐺 ∈ FriendGraph ∧ 1 < (β™―β€˜π΄) ∧ 1 < (β™―β€˜π΅)) β†’ βˆƒπ‘Ž ∈ 𝐴 βˆƒπ‘₯ ∈ 𝐴 βˆƒπ‘ ∈ 𝐡 βˆƒπ‘¦ ∈ 𝐡 ((π‘Ž β‰  π‘₯ ∧ 𝑏 β‰  𝑦) ∧ ({π‘Ž, 𝑏} ∈ (Edgβ€˜πΊ) ∧ {𝑏, π‘₯} ∈ (Edgβ€˜πΊ)) ∧ ({π‘₯, 𝑦} ∈ (Edgβ€˜πΊ) ∧ {𝑦, π‘Ž} ∈ (Edgβ€˜πΊ))))
30 frgrusgr 29247 . . . . . . . . . . . . . . . 16 (𝐺 ∈ FriendGraph β†’ 𝐺 ∈ USGraph)
31 simplll 774 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐺 ∈ USGraph ∧ (π‘Ž ∈ 𝐴 ∧ π‘₯ ∈ 𝐴)) ∧ (𝑏 ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) ∧ (π‘Ž β‰  π‘₯ ∧ 𝑏 β‰  𝑦)) β†’ 𝐺 ∈ USGraph)
32 elrabi 3644 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (π‘Ž ∈ {π‘₯ ∈ 𝑉 ∣ (π·β€˜π‘₯) = 𝐾} β†’ π‘Ž ∈ 𝑉)
3332, 3eleq2s 2856 . . . . . . . . . . . . . . . . . . . . . . . . 25 (π‘Ž ∈ 𝐴 β†’ π‘Ž ∈ 𝑉)
3433adantr 482 . . . . . . . . . . . . . . . . . . . . . . . 24 ((π‘Ž ∈ 𝐴 ∧ π‘₯ ∈ 𝐴) β†’ π‘Ž ∈ 𝑉)
3534ad3antlr 730 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐺 ∈ USGraph ∧ (π‘Ž ∈ 𝐴 ∧ π‘₯ ∈ 𝐴)) ∧ (𝑏 ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) ∧ (π‘Ž β‰  π‘₯ ∧ 𝑏 β‰  𝑦)) β†’ π‘Ž ∈ 𝑉)
36 rabidim1 3431 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (π‘₯ ∈ {π‘₯ ∈ 𝑉 ∣ (π·β€˜π‘₯) = 𝐾} β†’ π‘₯ ∈ 𝑉)
3736, 3eleq2s 2856 . . . . . . . . . . . . . . . . . . . . . . . . 25 (π‘₯ ∈ 𝐴 β†’ π‘₯ ∈ 𝑉)
3837adantl 483 . . . . . . . . . . . . . . . . . . . . . . . 24 ((π‘Ž ∈ 𝐴 ∧ π‘₯ ∈ 𝐴) β†’ π‘₯ ∈ 𝑉)
3938ad3antlr 730 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐺 ∈ USGraph ∧ (π‘Ž ∈ 𝐴 ∧ π‘₯ ∈ 𝐴)) ∧ (𝑏 ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) ∧ (π‘Ž β‰  π‘₯ ∧ 𝑏 β‰  𝑦)) β†’ π‘₯ ∈ 𝑉)
40 simprl 770 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐺 ∈ USGraph ∧ (π‘Ž ∈ 𝐴 ∧ π‘₯ ∈ 𝐴)) ∧ (𝑏 ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) ∧ (π‘Ž β‰  π‘₯ ∧ 𝑏 β‰  𝑦)) β†’ π‘Ž β‰  π‘₯)
41 eldifi 4091 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑏 ∈ (𝑉 βˆ– 𝐴) β†’ 𝑏 ∈ 𝑉)
4241, 4eleq2s 2856 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑏 ∈ 𝐡 β†’ 𝑏 ∈ 𝑉)
4342adantr 482 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑏 ∈ 𝐡 ∧ 𝑦 ∈ 𝐡) β†’ 𝑏 ∈ 𝑉)
4443ad2antlr 726 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐺 ∈ USGraph ∧ (π‘Ž ∈ 𝐴 ∧ π‘₯ ∈ 𝐴)) ∧ (𝑏 ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) ∧ (π‘Ž β‰  π‘₯ ∧ 𝑏 β‰  𝑦)) β†’ 𝑏 ∈ 𝑉)
45 eldifi 4091 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ (𝑉 βˆ– 𝐴) β†’ 𝑦 ∈ 𝑉)
4645, 4eleq2s 2856 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 ∈ 𝐡 β†’ 𝑦 ∈ 𝑉)
4746adantl 483 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑏 ∈ 𝐡 ∧ 𝑦 ∈ 𝐡) β†’ 𝑦 ∈ 𝑉)
4847ad2antlr 726 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐺 ∈ USGraph ∧ (π‘Ž ∈ 𝐴 ∧ π‘₯ ∈ 𝐴)) ∧ (𝑏 ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) ∧ (π‘Ž β‰  π‘₯ ∧ 𝑏 β‰  𝑦)) β†’ 𝑦 ∈ 𝑉)
49 simprr 772 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐺 ∈ USGraph ∧ (π‘Ž ∈ 𝐴 ∧ π‘₯ ∈ 𝐴)) ∧ (𝑏 ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) ∧ (π‘Ž β‰  π‘₯ ∧ 𝑏 β‰  𝑦)) β†’ 𝑏 β‰  𝑦)
501, 284cyclusnfrgr 29278 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐺 ∈ USGraph ∧ (π‘Ž ∈ 𝑉 ∧ π‘₯ ∈ 𝑉 ∧ π‘Ž β‰  π‘₯) ∧ (𝑏 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑏 β‰  𝑦)) β†’ ((({π‘Ž, 𝑏} ∈ (Edgβ€˜πΊ) ∧ {𝑏, π‘₯} ∈ (Edgβ€˜πΊ)) ∧ ({π‘₯, 𝑦} ∈ (Edgβ€˜πΊ) ∧ {𝑦, π‘Ž} ∈ (Edgβ€˜πΊ))) β†’ 𝐺 βˆ‰ FriendGraph ))
5131, 35, 39, 40, 44, 48, 49, 50syl133anc 1394 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐺 ∈ USGraph ∧ (π‘Ž ∈ 𝐴 ∧ π‘₯ ∈ 𝐴)) ∧ (𝑏 ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) ∧ (π‘Ž β‰  π‘₯ ∧ 𝑏 β‰  𝑦)) β†’ ((({π‘Ž, 𝑏} ∈ (Edgβ€˜πΊ) ∧ {𝑏, π‘₯} ∈ (Edgβ€˜πΊ)) ∧ ({π‘₯, 𝑦} ∈ (Edgβ€˜πΊ) ∧ {𝑦, π‘Ž} ∈ (Edgβ€˜πΊ))) β†’ 𝐺 βˆ‰ FriendGraph ))
5251exp4b 432 . . . . . . . . . . . . . . . . . . . . 21 (((𝐺 ∈ USGraph ∧ (π‘Ž ∈ 𝐴 ∧ π‘₯ ∈ 𝐴)) ∧ (𝑏 ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ ((π‘Ž β‰  π‘₯ ∧ 𝑏 β‰  𝑦) β†’ (({π‘Ž, 𝑏} ∈ (Edgβ€˜πΊ) ∧ {𝑏, π‘₯} ∈ (Edgβ€˜πΊ)) β†’ (({π‘₯, 𝑦} ∈ (Edgβ€˜πΊ) ∧ {𝑦, π‘Ž} ∈ (Edgβ€˜πΊ)) β†’ 𝐺 βˆ‰ FriendGraph ))))
53523impd 1349 . . . . . . . . . . . . . . . . . . . 20 (((𝐺 ∈ USGraph ∧ (π‘Ž ∈ 𝐴 ∧ π‘₯ ∈ 𝐴)) ∧ (𝑏 ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (((π‘Ž β‰  π‘₯ ∧ 𝑏 β‰  𝑦) ∧ ({π‘Ž, 𝑏} ∈ (Edgβ€˜πΊ) ∧ {𝑏, π‘₯} ∈ (Edgβ€˜πΊ)) ∧ ({π‘₯, 𝑦} ∈ (Edgβ€˜πΊ) ∧ {𝑦, π‘Ž} ∈ (Edgβ€˜πΊ))) β†’ 𝐺 βˆ‰ FriendGraph ))
54 df-nel 3051 . . . . . . . . . . . . . . . . . . . . 21 (𝐺 βˆ‰ FriendGraph ↔ Β¬ 𝐺 ∈ FriendGraph )
55 pm2.21 123 . . . . . . . . . . . . . . . . . . . . 21 (Β¬ 𝐺 ∈ FriendGraph β†’ (𝐺 ∈ FriendGraph β†’ (((β™―β€˜π΄) = 1 ∨ 𝐴 = βˆ…) ∨ ((β™―β€˜π΅) = 1 ∨ 𝐡 = βˆ…))))
5654, 55sylbi 216 . . . . . . . . . . . . . . . . . . . 20 (𝐺 βˆ‰ FriendGraph β†’ (𝐺 ∈ FriendGraph β†’ (((β™―β€˜π΄) = 1 ∨ 𝐴 = βˆ…) ∨ ((β™―β€˜π΅) = 1 ∨ 𝐡 = βˆ…))))
5753, 56syl6 35 . . . . . . . . . . . . . . . . . . 19 (((𝐺 ∈ USGraph ∧ (π‘Ž ∈ 𝐴 ∧ π‘₯ ∈ 𝐴)) ∧ (𝑏 ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (((π‘Ž β‰  π‘₯ ∧ 𝑏 β‰  𝑦) ∧ ({π‘Ž, 𝑏} ∈ (Edgβ€˜πΊ) ∧ {𝑏, π‘₯} ∈ (Edgβ€˜πΊ)) ∧ ({π‘₯, 𝑦} ∈ (Edgβ€˜πΊ) ∧ {𝑦, π‘Ž} ∈ (Edgβ€˜πΊ))) β†’ (𝐺 ∈ FriendGraph β†’ (((β™―β€˜π΄) = 1 ∨ 𝐴 = βˆ…) ∨ ((β™―β€˜π΅) = 1 ∨ 𝐡 = βˆ…)))))
5857rexlimdvva 3206 . . . . . . . . . . . . . . . . . 18 ((𝐺 ∈ USGraph ∧ (π‘Ž ∈ 𝐴 ∧ π‘₯ ∈ 𝐴)) β†’ (βˆƒπ‘ ∈ 𝐡 βˆƒπ‘¦ ∈ 𝐡 ((π‘Ž β‰  π‘₯ ∧ 𝑏 β‰  𝑦) ∧ ({π‘Ž, 𝑏} ∈ (Edgβ€˜πΊ) ∧ {𝑏, π‘₯} ∈ (Edgβ€˜πΊ)) ∧ ({π‘₯, 𝑦} ∈ (Edgβ€˜πΊ) ∧ {𝑦, π‘Ž} ∈ (Edgβ€˜πΊ))) β†’ (𝐺 ∈ FriendGraph β†’ (((β™―β€˜π΄) = 1 ∨ 𝐴 = βˆ…) ∨ ((β™―β€˜π΅) = 1 ∨ 𝐡 = βˆ…)))))
5958rexlimdvva 3206 . . . . . . . . . . . . . . . . 17 (𝐺 ∈ USGraph β†’ (βˆƒπ‘Ž ∈ 𝐴 βˆƒπ‘₯ ∈ 𝐴 βˆƒπ‘ ∈ 𝐡 βˆƒπ‘¦ ∈ 𝐡 ((π‘Ž β‰  π‘₯ ∧ 𝑏 β‰  𝑦) ∧ ({π‘Ž, 𝑏} ∈ (Edgβ€˜πΊ) ∧ {𝑏, π‘₯} ∈ (Edgβ€˜πΊ)) ∧ ({π‘₯, 𝑦} ∈ (Edgβ€˜πΊ) ∧ {𝑦, π‘Ž} ∈ (Edgβ€˜πΊ))) β†’ (𝐺 ∈ FriendGraph β†’ (((β™―β€˜π΄) = 1 ∨ 𝐴 = βˆ…) ∨ ((β™―β€˜π΅) = 1 ∨ 𝐡 = βˆ…)))))
6059com23 86 . . . . . . . . . . . . . . . 16 (𝐺 ∈ USGraph β†’ (𝐺 ∈ FriendGraph β†’ (βˆƒπ‘Ž ∈ 𝐴 βˆƒπ‘₯ ∈ 𝐴 βˆƒπ‘ ∈ 𝐡 βˆƒπ‘¦ ∈ 𝐡 ((π‘Ž β‰  π‘₯ ∧ 𝑏 β‰  𝑦) ∧ ({π‘Ž, 𝑏} ∈ (Edgβ€˜πΊ) ∧ {𝑏, π‘₯} ∈ (Edgβ€˜πΊ)) ∧ ({π‘₯, 𝑦} ∈ (Edgβ€˜πΊ) ∧ {𝑦, π‘Ž} ∈ (Edgβ€˜πΊ))) β†’ (((β™―β€˜π΄) = 1 ∨ 𝐴 = βˆ…) ∨ ((β™―β€˜π΅) = 1 ∨ 𝐡 = βˆ…)))))
6130, 60mpcom 38 . . . . . . . . . . . . . . 15 (𝐺 ∈ FriendGraph β†’ (βˆƒπ‘Ž ∈ 𝐴 βˆƒπ‘₯ ∈ 𝐴 βˆƒπ‘ ∈ 𝐡 βˆƒπ‘¦ ∈ 𝐡 ((π‘Ž β‰  π‘₯ ∧ 𝑏 β‰  𝑦) ∧ ({π‘Ž, 𝑏} ∈ (Edgβ€˜πΊ) ∧ {𝑏, π‘₯} ∈ (Edgβ€˜πΊ)) ∧ ({π‘₯, 𝑦} ∈ (Edgβ€˜πΊ) ∧ {𝑦, π‘Ž} ∈ (Edgβ€˜πΊ))) β†’ (((β™―β€˜π΄) = 1 ∨ 𝐴 = βˆ…) ∨ ((β™―β€˜π΅) = 1 ∨ 𝐡 = βˆ…))))
62613ad2ant1 1134 . . . . . . . . . . . . . 14 ((𝐺 ∈ FriendGraph ∧ 1 < (β™―β€˜π΄) ∧ 1 < (β™―β€˜π΅)) β†’ (βˆƒπ‘Ž ∈ 𝐴 βˆƒπ‘₯ ∈ 𝐴 βˆƒπ‘ ∈ 𝐡 βˆƒπ‘¦ ∈ 𝐡 ((π‘Ž β‰  π‘₯ ∧ 𝑏 β‰  𝑦) ∧ ({π‘Ž, 𝑏} ∈ (Edgβ€˜πΊ) ∧ {𝑏, π‘₯} ∈ (Edgβ€˜πΊ)) ∧ ({π‘₯, 𝑦} ∈ (Edgβ€˜πΊ) ∧ {𝑦, π‘Ž} ∈ (Edgβ€˜πΊ))) β†’ (((β™―β€˜π΄) = 1 ∨ 𝐴 = βˆ…) ∨ ((β™―β€˜π΅) = 1 ∨ 𝐡 = βˆ…))))
6329, 62mpd 15 . . . . . . . . . . . . 13 ((𝐺 ∈ FriendGraph ∧ 1 < (β™―β€˜π΄) ∧ 1 < (β™―β€˜π΅)) β†’ (((β™―β€˜π΄) = 1 ∨ 𝐴 = βˆ…) ∨ ((β™―β€˜π΅) = 1 ∨ 𝐡 = βˆ…)))
64633exp 1120 . . . . . . . . . . . 12 (𝐺 ∈ FriendGraph β†’ (1 < (β™―β€˜π΄) β†’ (1 < (β™―β€˜π΅) β†’ (((β™―β€˜π΄) = 1 ∨ 𝐴 = βˆ…) ∨ ((β™―β€˜π΅) = 1 ∨ 𝐡 = βˆ…)))))
6564com3l 89 . . . . . . . . . . 11 (1 < (β™―β€˜π΄) β†’ (1 < (β™―β€˜π΅) β†’ (𝐺 ∈ FriendGraph β†’ (((β™―β€˜π΄) = 1 ∨ 𝐴 = βˆ…) ∨ ((β™―β€˜π΅) = 1 ∨ 𝐡 = βˆ…)))))
6624, 27, 653jaoi 1428 . . . . . . . . . 10 ((𝐴 = βˆ… ∨ (β™―β€˜π΄) = 1 ∨ 1 < (β™―β€˜π΄)) β†’ (1 < (β™―β€˜π΅) β†’ (𝐺 ∈ FriendGraph β†’ (((β™―β€˜π΄) = 1 ∨ 𝐴 = βˆ…) ∨ ((β™―β€˜π΅) = 1 ∨ 𝐡 = βˆ…)))))
6766com12 32 . . . . . . . . 9 (1 < (β™―β€˜π΅) β†’ ((𝐴 = βˆ… ∨ (β™―β€˜π΄) = 1 ∨ 1 < (β™―β€˜π΄)) β†’ (𝐺 ∈ FriendGraph β†’ (((β™―β€˜π΄) = 1 ∨ 𝐴 = βˆ…) ∨ ((β™―β€˜π΅) = 1 ∨ 𝐡 = βˆ…)))))
6818, 21, 673jaoi 1428 . . . . . . . 8 ((𝐡 = βˆ… ∨ (β™―β€˜π΅) = 1 ∨ 1 < (β™―β€˜π΅)) β†’ ((𝐴 = βˆ… ∨ (β™―β€˜π΄) = 1 ∨ 1 < (β™―β€˜π΄)) β†’ (𝐺 ∈ FriendGraph β†’ (((β™―β€˜π΄) = 1 ∨ 𝐴 = βˆ…) ∨ ((β™―β€˜π΅) = 1 ∨ 𝐡 = βˆ…)))))
6915, 68syl6bi 253 . . . . . . 7 (𝐡 ∈ V β†’ (((β™―β€˜π΅) = 0 ∨ (β™―β€˜π΅) = 1 ∨ 1 < (β™―β€˜π΅)) β†’ ((𝐴 = βˆ… ∨ (β™―β€˜π΄) = 1 ∨ 1 < (β™―β€˜π΄)) β†’ (𝐺 ∈ FriendGraph β†’ (((β™―β€˜π΄) = 1 ∨ 𝐴 = βˆ…) ∨ ((β™―β€˜π΅) = 1 ∨ 𝐡 = βˆ…))))))
7011, 69mpd 15 . . . . . 6 (𝐡 ∈ V β†’ ((𝐴 = βˆ… ∨ (β™―β€˜π΄) = 1 ∨ 1 < (β™―β€˜π΄)) β†’ (𝐺 ∈ FriendGraph β†’ (((β™―β€˜π΄) = 1 ∨ 𝐴 = βˆ…) ∨ ((β™―β€˜π΅) = 1 ∨ 𝐡 = βˆ…)))))
7170com12 32 . . . . 5 ((𝐴 = βˆ… ∨ (β™―β€˜π΄) = 1 ∨ 1 < (β™―β€˜π΄)) β†’ (𝐡 ∈ V β†’ (𝐺 ∈ FriendGraph β†’ (((β™―β€˜π΄) = 1 ∨ 𝐴 = βˆ…) ∨ ((β™―β€˜π΅) = 1 ∨ 𝐡 = βˆ…)))))
7210, 71syl6bi 253 . . . 4 (𝐴 ∈ V β†’ (((β™―β€˜π΄) = 0 ∨ (β™―β€˜π΄) = 1 ∨ 1 < (β™―β€˜π΄)) β†’ (𝐡 ∈ V β†’ (𝐺 ∈ FriendGraph β†’ (((β™―β€˜π΄) = 1 ∨ 𝐴 = βˆ…) ∨ ((β™―β€˜π΅) = 1 ∨ 𝐡 = βˆ…))))))
736, 72mpd 15 . . 3 (𝐴 ∈ V β†’ (𝐡 ∈ V β†’ (𝐺 ∈ FriendGraph β†’ (((β™―β€˜π΄) = 1 ∨ 𝐴 = βˆ…) ∨ ((β™―β€˜π΅) = 1 ∨ 𝐡 = βˆ…)))))
7473imp 408 . 2 ((𝐴 ∈ V ∧ 𝐡 ∈ V) β†’ (𝐺 ∈ FriendGraph β†’ (((β™―β€˜π΄) = 1 ∨ 𝐴 = βˆ…) ∨ ((β™―β€˜π΅) = 1 ∨ 𝐡 = βˆ…))))
755, 74ax-mp 5 1 (𝐺 ∈ FriendGraph β†’ (((β™―β€˜π΄) = 1 ∨ 𝐴 = βˆ…) ∨ ((β™―β€˜π΅) = 1 ∨ 𝐡 = βˆ…)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 397   ∨ wo 846   ∨ w3o 1087   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2944   βˆ‰ wnel 3050  βˆƒwrex 3074  {crab 3410  Vcvv 3448   βˆ– cdif 3912  βˆ…c0 4287  {cpr 4593   class class class wbr 5110  β€˜cfv 6501  0cc0 11058  1c1 11059   < clt 11196  β™―chash 14237  Vtxcvtx 27989  Edgcedg 28040  USGraphcusgr 28142  VtxDegcvtxdg 28455   FriendGraph cfrgr 29244
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-2o 8418  df-oadd 8421  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-dju 9844  df-card 9882  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-nn 12161  df-2 12223  df-n0 12421  df-xnn0 12493  df-z 12507  df-uz 12771  df-xadd 13041  df-fz 13432  df-hash 14238  df-edg 28041  df-uhgr 28051  df-ushgr 28052  df-upgr 28075  df-umgr 28076  df-uspgr 28143  df-usgr 28144  df-nbgr 28323  df-vtxdg 28456  df-frgr 29245
This theorem is referenced by:  frgrregorufr0  29310
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