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Theorem 1vwmgr 27625
Description: Every graph with one vertex (which may be connect with itself by (multiple) loops!) is a windmill graph. (Contributed by Alexander van der Vekens, 5-Oct-2017.) (Revised by AV, 31-Mar-2021.)
Assertion
Ref Expression
1vwmgr ((𝐴𝑋𝑉 = {𝐴}) → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸))
Distinct variable groups:   𝐴,,𝑣,𝑤   ,𝐸   ,𝑉,𝑣,𝑤
Allowed substitution hints:   𝐸(𝑤,𝑣)   𝑋(𝑤,𝑣,)

Proof of Theorem 1vwmgr
StepHypRef Expression
1 ral0 4269 . . . 4 𝑣 ∈ ∅ ({𝑣, 𝐴} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {𝐴}){𝑣, 𝑤} ∈ 𝐸)
2 sneq 4378 . . . . . . . 8 ( = 𝐴 → {} = {𝐴})
32difeq2d 3926 . . . . . . 7 ( = 𝐴 → ({𝐴} ∖ {}) = ({𝐴} ∖ {𝐴}))
4 difid 4149 . . . . . . 7 ({𝐴} ∖ {𝐴}) = ∅
53, 4syl6eq 2849 . . . . . 6 ( = 𝐴 → ({𝐴} ∖ {}) = ∅)
6 preq2 4458 . . . . . . . 8 ( = 𝐴 → {𝑣, } = {𝑣, 𝐴})
76eleq1d 2863 . . . . . . 7 ( = 𝐴 → ({𝑣, } ∈ 𝐸 ↔ {𝑣, 𝐴} ∈ 𝐸))
8 reueq1 3323 . . . . . . . 8 (({𝐴} ∖ {}) = ({𝐴} ∖ {𝐴}) → (∃!𝑤 ∈ ({𝐴} ∖ {}){𝑣, 𝑤} ∈ 𝐸 ↔ ∃!𝑤 ∈ ({𝐴} ∖ {𝐴}){𝑣, 𝑤} ∈ 𝐸))
93, 8syl 17 . . . . . . 7 ( = 𝐴 → (∃!𝑤 ∈ ({𝐴} ∖ {}){𝑣, 𝑤} ∈ 𝐸 ↔ ∃!𝑤 ∈ ({𝐴} ∖ {𝐴}){𝑣, 𝑤} ∈ 𝐸))
107, 9anbi12d 625 . . . . . 6 ( = 𝐴 → (({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {}){𝑣, 𝑤} ∈ 𝐸) ↔ ({𝑣, 𝐴} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {𝐴}){𝑣, 𝑤} ∈ 𝐸)))
115, 10raleqbidv 3335 . . . . 5 ( = 𝐴 → (∀𝑣 ∈ ({𝐴} ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {}){𝑣, 𝑤} ∈ 𝐸) ↔ ∀𝑣 ∈ ∅ ({𝑣, 𝐴} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {𝐴}){𝑣, 𝑤} ∈ 𝐸)))
1211rexsng 4410 . . . 4 (𝐴𝑋 → (∃ ∈ {𝐴}∀𝑣 ∈ ({𝐴} ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {}){𝑣, 𝑤} ∈ 𝐸) ↔ ∀𝑣 ∈ ∅ ({𝑣, 𝐴} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {𝐴}){𝑣, 𝑤} ∈ 𝐸)))
131, 12mpbiri 250 . . 3 (𝐴𝑋 → ∃ ∈ {𝐴}∀𝑣 ∈ ({𝐴} ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {}){𝑣, 𝑤} ∈ 𝐸))
1413adantr 473 . 2 ((𝐴𝑋𝑉 = {𝐴}) → ∃ ∈ {𝐴}∀𝑣 ∈ ({𝐴} ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {}){𝑣, 𝑤} ∈ 𝐸))
15 difeq1 3919 . . . . 5 (𝑉 = {𝐴} → (𝑉 ∖ {}) = ({𝐴} ∖ {}))
16 reueq1 3323 . . . . . . 7 ((𝑉 ∖ {}) = ({𝐴} ∖ {}) → (∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸 ↔ ∃!𝑤 ∈ ({𝐴} ∖ {}){𝑣, 𝑤} ∈ 𝐸))
1715, 16syl 17 . . . . . 6 (𝑉 = {𝐴} → (∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸 ↔ ∃!𝑤 ∈ ({𝐴} ∖ {}){𝑣, 𝑤} ∈ 𝐸))
1817anbi2d 623 . . . . 5 (𝑉 = {𝐴} → (({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸) ↔ ({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {}){𝑣, 𝑤} ∈ 𝐸)))
1915, 18raleqbidv 3335 . . . 4 (𝑉 = {𝐴} → (∀𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸) ↔ ∀𝑣 ∈ ({𝐴} ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {}){𝑣, 𝑤} ∈ 𝐸)))
2019rexeqbi1dv 3330 . . 3 (𝑉 = {𝐴} → (∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸) ↔ ∃ ∈ {𝐴}∀𝑣 ∈ ({𝐴} ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {}){𝑣, 𝑤} ∈ 𝐸)))
2120adantl 474 . 2 ((𝐴𝑋𝑉 = {𝐴}) → (∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸) ↔ ∃ ∈ {𝐴}∀𝑣 ∈ ({𝐴} ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {}){𝑣, 𝑤} ∈ 𝐸)))
2214, 21mpbird 249 1 ((𝐴𝑋𝑉 = {𝐴}) → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  wral 3089  wrex 3090  ∃!wreu 3091  cdif 3766  c0 4115  {csn 4368  {cpr 4370
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-sn 4369  df-pr 4371
This theorem is referenced by:  1to2vfriswmgr  27628
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