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Theorem 1vwmgr 30310
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 4536 . . . 4 𝑣 ∈ ∅ ({𝑣, 𝐴} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {𝐴}){𝑣, 𝑤} ∈ 𝐸)
2 sneq 4658 . . . . . . . 8 ( = 𝐴 → {} = {𝐴})
32difeq2d 4149 . . . . . . 7 ( = 𝐴 → ({𝐴} ∖ {}) = ({𝐴} ∖ {𝐴}))
4 difid 4398 . . . . . . 7 ({𝐴} ∖ {𝐴}) = ∅
53, 4eqtrdi 2796 . . . . . 6 ( = 𝐴 → ({𝐴} ∖ {}) = ∅)
6 preq2 4759 . . . . . . . 8 ( = 𝐴 → {𝑣, } = {𝑣, 𝐴})
76eleq1d 2829 . . . . . . 7 ( = 𝐴 → ({𝑣, } ∈ 𝐸 ↔ {𝑣, 𝐴} ∈ 𝐸))
8 reueq1 3426 . . . . . . . 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 631 . . . . . 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 4698 . . . 4 (𝐴𝑋 → (∃ ∈ {𝐴}∀𝑣 ∈ ({𝐴} ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {}){𝑣, 𝑤} ∈ 𝐸) ↔ ∀𝑣 ∈ ∅ ({𝑣, 𝐴} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {𝐴}){𝑣, 𝑤} ∈ 𝐸)))
131, 12mpbiri 258 . . 3 (𝐴𝑋 → ∃ ∈ {𝐴}∀𝑣 ∈ ({𝐴} ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {}){𝑣, 𝑤} ∈ 𝐸))
1413adantr 480 . 2 ((𝐴𝑋𝑉 = {𝐴}) → ∃ ∈ {𝐴}∀𝑣 ∈ ({𝐴} ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {}){𝑣, 𝑤} ∈ 𝐸))
15 difeq1 4142 . . . . 5 (𝑉 = {𝐴} → (𝑉 ∖ {}) = ({𝐴} ∖ {}))
16 reueq1 3426 . . . . . . 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 629 . . . . 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 3347 . . 3 (𝑉 = {𝐴} → (∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸) ↔ ∃ ∈ {𝐴}∀𝑣 ∈ ({𝐴} ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {}){𝑣, 𝑤} ∈ 𝐸)))
2120adantl 481 . 2 ((𝐴𝑋𝑉 = {𝐴}) → (∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸) ↔ ∃ ∈ {𝐴}∀𝑣 ∈ ({𝐴} ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {}){𝑣, 𝑤} ∈ 𝐸)))
2214, 21mpbird 257 1 ((𝐴𝑋𝑉 = {𝐴}) → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2108  wral 3067  wrex 3076  ∃!wreu 3386  cdif 3973  c0 4352  {csn 4648  {cpr 4650
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-nul 4353  df-sn 4649  df-pr 4651
This theorem is referenced by:  1to2vfriswmgr  30313
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