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Theorem 2pthfrgrrn 27464
Description: Between any two (different) vertices in a friendship graph is a 2-path (path of length 2), see Proposition 1(b) of [MertziosUnger] p. 153 : "A friendship graph G ..., as well as the distance between any two nodes in G is at most two". (Contributed by Alexander van der Vekens, 15-Nov-2017.) (Revised by AV, 1-Apr-2021.)
Hypotheses
Ref Expression
2pthfrgrrn.v 𝑉 = (Vtx‘𝐺)
2pthfrgrrn.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
2pthfrgrrn (𝐺 ∈ FriendGraph → ∀𝑎𝑉𝑐 ∈ (𝑉 ∖ {𝑎})∃𝑏𝑉 ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸))
Distinct variable groups:   𝐺,𝑎,𝑏,𝑐   𝑉,𝑎,𝑏,𝑐
Allowed substitution hints:   𝐸(𝑎,𝑏,𝑐)

Proof of Theorem 2pthfrgrrn
StepHypRef Expression
1 2pthfrgrrn.v . . 3 𝑉 = (Vtx‘𝐺)
2 2pthfrgrrn.e . . 3 𝐸 = (Edg‘𝐺)
31, 2frgrusgrfrcond 27441 . 2 (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑎𝑉𝑐 ∈ (𝑉 ∖ {𝑎})∃!𝑏𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ 𝐸))
4 reurex 3309 . . . . . 6 (∃!𝑏𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ 𝐸 → ∃𝑏𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ 𝐸)
5 prcom 4404 . . . . . . . . . 10 {𝑎, 𝑏} = {𝑏, 𝑎}
65eleq1i 2841 . . . . . . . . 9 ({𝑎, 𝑏} ∈ 𝐸 ↔ {𝑏, 𝑎} ∈ 𝐸)
76anbi1i 610 . . . . . . . 8 (({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸) ↔ ({𝑏, 𝑎} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸))
8 zfpair2 5036 . . . . . . . . 9 {𝑏, 𝑎} ∈ V
9 zfpair2 5036 . . . . . . . . 9 {𝑏, 𝑐} ∈ V
108, 9prss 4487 . . . . . . . 8 (({𝑏, 𝑎} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸) ↔ {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ 𝐸)
117, 10sylbbr 226 . . . . . . 7 ({{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ 𝐸 → ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸))
1211reximi 3159 . . . . . 6 (∃𝑏𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ 𝐸 → ∃𝑏𝑉 ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸))
134, 12syl 17 . . . . 5 (∃!𝑏𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ 𝐸 → ∃𝑏𝑉 ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸))
1413a1i 11 . . . 4 ((𝐺 ∈ USGraph ∧ (𝑎𝑉𝑐 ∈ (𝑉 ∖ {𝑎}))) → (∃!𝑏𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ 𝐸 → ∃𝑏𝑉 ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸)))
1514ralimdvva 3113 . . 3 (𝐺 ∈ USGraph → (∀𝑎𝑉𝑐 ∈ (𝑉 ∖ {𝑎})∃!𝑏𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ 𝐸 → ∀𝑎𝑉𝑐 ∈ (𝑉 ∖ {𝑎})∃𝑏𝑉 ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸)))
1615imp 393 . 2 ((𝐺 ∈ USGraph ∧ ∀𝑎𝑉𝑐 ∈ (𝑉 ∖ {𝑎})∃!𝑏𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ 𝐸) → ∀𝑎𝑉𝑐 ∈ (𝑉 ∖ {𝑎})∃𝑏𝑉 ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸))
173, 16sylbi 207 1 (𝐺 ∈ FriendGraph → ∀𝑎𝑉𝑐 ∈ (𝑉 ∖ {𝑎})∃𝑏𝑉 ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wral 3061  wrex 3062  ∃!wreu 3063  cdif 3720  wss 3723  {csn 4317  {cpr 4319  cfv 6030  Vtxcvtx 26095  Edgcedg 26160  USGraphcusgr 26266   FriendGraph cfrgr 27438
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-iota 5993  df-fv 6038  df-frgr 27439
This theorem is referenced by:  2pthfrgrrn2  27465  3cyclfrgrrn1  27467
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