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Theorem 1vwmgr 28102
 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 4416 . . . 4 𝑣 ∈ ∅ ({𝑣, 𝐴} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {𝐴}){𝑣, 𝑤} ∈ 𝐸)
2 sneq 4537 . . . . . . . 8 ( = 𝐴 → {} = {𝐴})
32difeq2d 4052 . . . . . . 7 ( = 𝐴 → ({𝐴} ∖ {}) = ({𝐴} ∖ {𝐴}))
4 difid 4286 . . . . . . 7 ({𝐴} ∖ {𝐴}) = ∅
53, 4eqtrdi 2849 . . . . . 6 ( = 𝐴 → ({𝐴} ∖ {}) = ∅)
6 preq2 4632 . . . . . . . 8 ( = 𝐴 → {𝑣, } = {𝑣, 𝐴})
76eleq1d 2874 . . . . . . 7 ( = 𝐴 → ({𝑣, } ∈ 𝐸 ↔ {𝑣, 𝐴} ∈ 𝐸))
8 reueq1 3360 . . . . . . . 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 633 . . . . . 6 ( = 𝐴 → (({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {}){𝑣, 𝑤} ∈ 𝐸) ↔ ({𝑣, 𝐴} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {𝐴}){𝑣, 𝑤} ∈ 𝐸)))
115, 10raleqbidv 3354 . . . . 5 ( = 𝐴 → (∀𝑣 ∈ ({𝐴} ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {}){𝑣, 𝑤} ∈ 𝐸) ↔ ∀𝑣 ∈ ∅ ({𝑣, 𝐴} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {𝐴}){𝑣, 𝑤} ∈ 𝐸)))
1211rexsng 4576 . . . 4 (𝐴𝑋 → (∃ ∈ {𝐴}∀𝑣 ∈ ({𝐴} ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {}){𝑣, 𝑤} ∈ 𝐸) ↔ ∀𝑣 ∈ ∅ ({𝑣, 𝐴} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {𝐴}){𝑣, 𝑤} ∈ 𝐸)))
131, 12mpbiri 261 . . 3 (𝐴𝑋 → ∃ ∈ {𝐴}∀𝑣 ∈ ({𝐴} ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {}){𝑣, 𝑤} ∈ 𝐸))
1413adantr 484 . 2 ((𝐴𝑋𝑉 = {𝐴}) → ∃ ∈ {𝐴}∀𝑣 ∈ ({𝐴} ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {}){𝑣, 𝑤} ∈ 𝐸))
15 difeq1 4045 . . . . 5 (𝑉 = {𝐴} → (𝑉 ∖ {}) = ({𝐴} ∖ {}))
16 reueq1 3360 . . . . . . 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 631 . . . . 5 (𝑉 = {𝐴} → (({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸) ↔ ({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {}){𝑣, 𝑤} ∈ 𝐸)))
1915, 18raleqbidv 3354 . . . 4 (𝑉 = {𝐴} → (∀𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸) ↔ ∀𝑣 ∈ ({𝐴} ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {}){𝑣, 𝑤} ∈ 𝐸)))
2019rexeqbi1dv 3357 . . 3 (𝑉 = {𝐴} → (∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸) ↔ ∃ ∈ {𝐴}∀𝑣 ∈ ({𝐴} ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {}){𝑣, 𝑤} ∈ 𝐸)))
2120adantl 485 . 2 ((𝐴𝑋𝑉 = {𝐴}) → (∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸) ↔ ∃ ∈ {𝐴}∀𝑣 ∈ ({𝐴} ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {}){𝑣, 𝑤} ∈ 𝐸)))
2214, 21mpbird 260 1 ((𝐴𝑋𝑉 = {𝐴}) → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107  ∃!wreu 3108   ∖ cdif 3879  ∅c0 4245  {csn 4527  {cpr 4529 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3722  df-dif 3885  df-un 3887  df-in 3889  df-ss 3899  df-nul 4246  df-sn 4528  df-pr 4530 This theorem is referenced by:  1to2vfriswmgr  28105
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