Theorem frgraregorufr 26580

Description: If there is a vertex having degree 𝐾 for each (nonnegative integer) 𝐾 in a friendship graph, then either all vertices have degree 𝐾 or there is a universal friend. This corresponds to claim 2 in [Huneke] p. 2: "Suppose there is a vertex of degree k > 1. ... all vertices have degree k, unless there is a universal friend. ... It follows that G is k-regular, i.e., the degree of every vertex is k". (Contributed by Alexander van der Vekens, 1-Jan-2018.)

Assertion

Ref	Expression
frgraregorufr	⊢ (𝑉 FriendGrph 𝐸 → (∃𝑎 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑎) = 𝐾 → (∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)))

Distinct variable groups:   𝑣,𝑎,𝑤,𝐾   𝑉,𝑎,𝑣,𝑤   𝐸,𝑎,𝑣,𝑤

Proof of Theorem frgraregorufr

Step	Hyp	Ref	Expression
1		frgraregorufr0 26579	. 2 ⊢ (𝑉 FriendGrph 𝐸 → (∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 ∨ ∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) ≠ 𝐾 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸))
2		orc 399	. . . 4 ⊢ (∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 → (∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸))
3	2	a1d 25	. . 3 ⊢ (∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 → (∃𝑎 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑎) = 𝐾 → (∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)))
4		fveq2 6103	. . . . . . . 8 ⊢ (𝑣 = 𝑎 → ((𝑉 VDeg 𝐸)‘𝑣) = ((𝑉 VDeg 𝐸)‘𝑎))
5	4	neeq1d 2841	. . . . . . 7 ⊢ (𝑣 = 𝑎 → (((𝑉 VDeg 𝐸)‘𝑣) ≠ 𝐾 ↔ ((𝑉 VDeg 𝐸)‘𝑎) ≠ 𝐾))
6	5	rspcva 3280	. . . . . 6 ⊢ ((𝑎 ∈ 𝑉 ∧ ∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) ≠ 𝐾) → ((𝑉 VDeg 𝐸)‘𝑎) ≠ 𝐾)
7		df-ne 2782	. . . . . . 7 ⊢ (((𝑉 VDeg 𝐸)‘𝑎) ≠ 𝐾 ↔ ¬ ((𝑉 VDeg 𝐸)‘𝑎) = 𝐾)
8		pm2.21 119	. . . . . . 7 ⊢ (¬ ((𝑉 VDeg 𝐸)‘𝑎) = 𝐾 → (((𝑉 VDeg 𝐸)‘𝑎) = 𝐾 → (∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)))
9	7, 8	sylbi 206	. . . . . 6 ⊢ (((𝑉 VDeg 𝐸)‘𝑎) ≠ 𝐾 → (((𝑉 VDeg 𝐸)‘𝑎) = 𝐾 → (∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)))
10	6, 9	syl 17	. . . . 5 ⊢ ((𝑎 ∈ 𝑉 ∧ ∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) ≠ 𝐾) → (((𝑉 VDeg 𝐸)‘𝑎) = 𝐾 → (∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)))
11	10	ancoms 468	. . . 4 ⊢ ((∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) ≠ 𝐾 ∧ 𝑎 ∈ 𝑉) → (((𝑉 VDeg 𝐸)‘𝑎) = 𝐾 → (∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)))
12	11	rexlimdva 3013	. . 3 ⊢ (∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) ≠ 𝐾 → (∃𝑎 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑎) = 𝐾 → (∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)))
13		olc 398	. . . 4 ⊢ (∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸 → (∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸))
14	13	a1d 25	. . 3 ⊢ (∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸 → (∃𝑎 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑎) = 𝐾 → (∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)))
15	3, 12, 14	3jaoi 1383	. 2 ⊢ ((∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 ∨ ∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) ≠ 𝐾 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸) → (∃𝑎 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑎) = 𝐾 → (∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)))
16	1, 15	syl 17	1 ⊢ (𝑉 FriendGrph 𝐸 → (∃𝑎 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑎) = 𝐾 → (∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   ∧ wa 383   ∨ w3o 1030   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897   ∖ cdif 3537  {csn 4125  {cpr 4127   class class class wbr 4583  ran crn 5039  ‘cfv 5804  (class class class)co 6549   VDeg cvdg 26420   FriendGrph cfrgra 26515

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-xadd 11823  df-fz 12198  df-hash 12980  df-usgra 25862  df-nbgra 25949  df-vdgr 26421  df-frgra 26516

This theorem is referenced by:  frgraregorufrg  26599