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Theorem frgrwopreg 29614
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 29603 . 2 (𝐴 ∈ V ∧ 𝐡 ∈ V)
6 hashv01gt1 14307 . . . 4 (𝐴 ∈ V β†’ ((β™―β€˜π΄) = 0 ∨ (β™―β€˜π΄) = 1 ∨ 1 < (β™―β€˜π΄)))
7 hasheq0 14325 . . . . . 6 (𝐴 ∈ V β†’ ((β™―β€˜π΄) = 0 ↔ 𝐴 = βˆ…))
8 biidd 261 . . . . . 6 (𝐴 ∈ V β†’ ((β™―β€˜π΄) = 1 ↔ (β™―β€˜π΄) = 1))
9 biidd 261 . . . . . 6 (𝐴 ∈ V β†’ (1 < (β™―β€˜π΄) ↔ 1 < (β™―β€˜π΄)))
107, 8, 93orbi123d 1435 . . . . 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 14307 . . . . . . 7 (𝐡 ∈ V β†’ ((β™―β€˜π΅) = 0 ∨ (β™―β€˜π΅) = 1 ∨ 1 < (β™―β€˜π΅)))
12 hasheq0 14325 . . . . . . . . 9 (𝐡 ∈ V β†’ ((β™―β€˜π΅) = 0 ↔ 𝐡 = βˆ…))
13 biidd 261 . . . . . . . . 9 (𝐡 ∈ V β†’ ((β™―β€˜π΅) = 1 ↔ (β™―β€˜π΅) = 1))
14 biidd 261 . . . . . . . . 9 (𝐡 ∈ V β†’ (1 < (β™―β€˜π΅) ↔ 1 < (β™―β€˜π΅)))
1512, 13, 143orbi123d 1435 . . . . . . . 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 866 . . . . . . . . . . 11 (𝐡 = βˆ… β†’ ((β™―β€˜π΅) = 1 ∨ 𝐡 = βˆ…))
1716olcd 872 . . . . . . . . . 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 865 . . . . . . . . . . 11 ((β™―β€˜π΅) = 1 β†’ ((β™―β€˜π΅) = 1 ∨ 𝐡 = βˆ…))
2019olcd 872 . . . . . . . . . 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 866 . . . . . . . . . . . . 13 (𝐴 = βˆ… β†’ ((β™―β€˜π΄) = 1 ∨ 𝐴 = βˆ…))
2322orcd 871 . . . . . . . . . . . 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 865 . . . . . . . . . . . . 13 ((β™―β€˜π΄) = 1 β†’ ((β™―β€˜π΄) = 1 ∨ 𝐴 = βˆ…))
2625orcd 871 . . . . . . . . . . . 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 2732 . . . . . . . . . . . . . . 15 (Edgβ€˜πΊ) = (Edgβ€˜πΊ)
291, 2, 3, 4, 28frgrwopreglem5 29612 . . . . . . . . . . . . . 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 29552 . . . . . . . . . . . . . . . 16 (𝐺 ∈ FriendGraph β†’ 𝐺 ∈ USGraph)
31 simplll 773 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐺 ∈ USGraph ∧ (π‘Ž ∈ 𝐴 ∧ π‘₯ ∈ 𝐴)) ∧ (𝑏 ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) ∧ (π‘Ž β‰  π‘₯ ∧ 𝑏 β‰  𝑦)) β†’ 𝐺 ∈ USGraph)
32 elrabi 3677 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (π‘Ž ∈ {π‘₯ ∈ 𝑉 ∣ (π·β€˜π‘₯) = 𝐾} β†’ π‘Ž ∈ 𝑉)
3332, 3eleq2s 2851 . . . . . . . . . . . . . . . . . . . . . . . . 25 (π‘Ž ∈ 𝐴 β†’ π‘Ž ∈ 𝑉)
3433adantr 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((π‘Ž ∈ 𝐴 ∧ π‘₯ ∈ 𝐴) β†’ π‘Ž ∈ 𝑉)
3534ad3antlr 729 . . . . . . . . . . . . . . . . . . . . . . 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 3453 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (π‘₯ ∈ {π‘₯ ∈ 𝑉 ∣ (π·β€˜π‘₯) = 𝐾} β†’ π‘₯ ∈ 𝑉)
3736, 3eleq2s 2851 . . . . . . . . . . . . . . . . . . . . . . . . 25 (π‘₯ ∈ 𝐴 β†’ π‘₯ ∈ 𝑉)
3837adantl 482 . . . . . . . . . . . . . . . . . . . . . . . 24 ((π‘Ž ∈ 𝐴 ∧ π‘₯ ∈ 𝐴) β†’ π‘₯ ∈ 𝑉)
3938ad3antlr 729 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐺 ∈ USGraph ∧ (π‘Ž ∈ 𝐴 ∧ π‘₯ ∈ 𝐴)) ∧ (𝑏 ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) ∧ (π‘Ž β‰  π‘₯ ∧ 𝑏 β‰  𝑦)) β†’ π‘₯ ∈ 𝑉)
40 simprl 769 . . . . . . . . . . . . . . . . . . . . . . 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 4126 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑏 ∈ (𝑉 βˆ– 𝐴) β†’ 𝑏 ∈ 𝑉)
4241, 4eleq2s 2851 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑏 ∈ 𝐡 β†’ 𝑏 ∈ 𝑉)
4342adantr 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑏 ∈ 𝐡 ∧ 𝑦 ∈ 𝐡) β†’ 𝑏 ∈ 𝑉)
4443ad2antlr 725 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐺 ∈ USGraph ∧ (π‘Ž ∈ 𝐴 ∧ π‘₯ ∈ 𝐴)) ∧ (𝑏 ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) ∧ (π‘Ž β‰  π‘₯ ∧ 𝑏 β‰  𝑦)) β†’ 𝑏 ∈ 𝑉)
45 eldifi 4126 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ (𝑉 βˆ– 𝐴) β†’ 𝑦 ∈ 𝑉)
4645, 4eleq2s 2851 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 ∈ 𝐡 β†’ 𝑦 ∈ 𝑉)
4746adantl 482 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑏 ∈ 𝐡 ∧ 𝑦 ∈ 𝐡) β†’ 𝑦 ∈ 𝑉)
4847ad2antlr 725 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐺 ∈ USGraph ∧ (π‘Ž ∈ 𝐴 ∧ π‘₯ ∈ 𝐴)) ∧ (𝑏 ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) ∧ (π‘Ž β‰  π‘₯ ∧ 𝑏 β‰  𝑦)) β†’ 𝑦 ∈ 𝑉)
49 simprr 771 . . . . . . . . . . . . . . . . . . . . . . 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 29583 . . . . . . . . . . . . . . . . . . . . . . 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 1393 . . . . . . . . . . . . . . . . . . . . . 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 431 . . . . . . . . . . . . . . . . . . . . 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 1348 . . . . . . . . . . . . . . . . . . . 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 3047 . . . . . . . . . . . . . . . . . . . . 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 3211 . . . . . . . . . . . . . . . . . 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 3211 . . . . . . . . . . . . . . . . 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 1133 . . . . . . . . . . . . . 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 1119 . . . . . . . . . . . 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 1427 . . . . . . . . . 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 1427 . . . . . . . 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 252 . . . . . . 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 252 . . . 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 407 . 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 396   ∨ wo 845   ∨ w3o 1086   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   β‰  wne 2940   βˆ‰ wnel 3046  βˆƒwrex 3070  {crab 3432  Vcvv 3474   βˆ– cdif 3945  βˆ…c0 4322  {cpr 4630   class class class wbr 5148  β€˜cfv 6543  0cc0 11112  1c1 11113   < clt 11250  β™―chash 14292  Vtxcvtx 28294  Edgcedg 28345  USGraphcusgr 28447  VtxDegcvtxdg 28760   FriendGraph cfrgr 29549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-2o 8469  df-oadd 8472  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-dju 9898  df-card 9936  df-pnf 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-nn 12215  df-2 12277  df-n0 12475  df-xnn0 12547  df-z 12561  df-uz 12825  df-xadd 13095  df-fz 13487  df-hash 14293  df-edg 28346  df-uhgr 28356  df-ushgr 28357  df-upgr 28380  df-umgr 28381  df-uspgr 28448  df-usgr 28449  df-nbgr 28628  df-vtxdg 28761  df-frgr 29550
This theorem is referenced by:  frgrregorufr0  29615
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