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Theorem 1vwmgr 27004
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 4048 . . . 4 𝑣 ∈ ∅ ({𝑣, 𝐴} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {𝐴}){𝑣, 𝑤} ∈ 𝐸)
2 sneq 4158 . . . . . . . 8 ( = 𝐴 → {} = {𝐴})
32difeq2d 3706 . . . . . . 7 ( = 𝐴 → ({𝐴} ∖ {}) = ({𝐴} ∖ {𝐴}))
4 difid 3922 . . . . . . 7 ({𝐴} ∖ {𝐴}) = ∅
53, 4syl6eq 2671 . . . . . 6 ( = 𝐴 → ({𝐴} ∖ {}) = ∅)
6 preq2 4239 . . . . . . . 8 ( = 𝐴 → {𝑣, } = {𝑣, 𝐴})
76eleq1d 2683 . . . . . . 7 ( = 𝐴 → ({𝑣, } ∈ 𝐸 ↔ {𝑣, 𝐴} ∈ 𝐸))
8 reueq1 3129 . . . . . . . 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 746 . . . . . 6 ( = 𝐴 → (({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {}){𝑣, 𝑤} ∈ 𝐸) ↔ ({𝑣, 𝐴} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {𝐴}){𝑣, 𝑤} ∈ 𝐸)))
115, 10raleqbidv 3141 . . . . 5 ( = 𝐴 → (∀𝑣 ∈ ({𝐴} ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {}){𝑣, 𝑤} ∈ 𝐸) ↔ ∀𝑣 ∈ ∅ ({𝑣, 𝐴} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {𝐴}){𝑣, 𝑤} ∈ 𝐸)))
1211rexsng 4190 . . . 4 (𝐴𝑋 → (∃ ∈ {𝐴}∀𝑣 ∈ ({𝐴} ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {}){𝑣, 𝑤} ∈ 𝐸) ↔ ∀𝑣 ∈ ∅ ({𝑣, 𝐴} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {𝐴}){𝑣, 𝑤} ∈ 𝐸)))
131, 12mpbiri 248 . . 3 (𝐴𝑋 → ∃ ∈ {𝐴}∀𝑣 ∈ ({𝐴} ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {}){𝑣, 𝑤} ∈ 𝐸))
1413adantr 481 . 2 ((𝐴𝑋𝑉 = {𝐴}) → ∃ ∈ {𝐴}∀𝑣 ∈ ({𝐴} ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {}){𝑣, 𝑤} ∈ 𝐸))
15 difeq1 3699 . . . . 5 (𝑉 = {𝐴} → (𝑉 ∖ {}) = ({𝐴} ∖ {}))
16 reueq1 3129 . . . . . . 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 739 . . . . 5 (𝑉 = {𝐴} → (({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸) ↔ ({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {}){𝑣, 𝑤} ∈ 𝐸)))
1915, 18raleqbidv 3141 . . . 4 (𝑉 = {𝐴} → (∀𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸) ↔ ∀𝑣 ∈ ({𝐴} ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {}){𝑣, 𝑤} ∈ 𝐸)))
2019rexeqbi1dv 3136 . . 3 (𝑉 = {𝐴} → (∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸) ↔ ∃ ∈ {𝐴}∀𝑣 ∈ ({𝐴} ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {}){𝑣, 𝑤} ∈ 𝐸)))
2120adantl 482 . 2 ((𝐴𝑋𝑉 = {𝐴}) → (∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸) ↔ ∃ ∈ {𝐴}∀𝑣 ∈ ({𝐴} ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {}){𝑣, 𝑤} ∈ 𝐸)))
2214, 21mpbird 247 1 ((𝐴𝑋𝑉 = {𝐴}) → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  wrex 2908  ∃!wreu 2909  cdif 3552  c0 3891  {csn 4148  {cpr 4150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-sn 4149  df-pr 4151
This theorem is referenced by:  1to2vfriswmgr  27007
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