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Theorem 1vwmgr 27458
 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 4218 . . . 4 𝑣 ∈ ∅ ({𝑣, 𝐴} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {𝐴}){𝑣, 𝑤} ∈ 𝐸)
2 sneq 4327 . . . . . . . 8 ( = 𝐴 → {} = {𝐴})
32difeq2d 3879 . . . . . . 7 ( = 𝐴 → ({𝐴} ∖ {}) = ({𝐴} ∖ {𝐴}))
4 difid 4096 . . . . . . 7 ({𝐴} ∖ {𝐴}) = ∅
53, 4syl6eq 2821 . . . . . 6 ( = 𝐴 → ({𝐴} ∖ {}) = ∅)
6 preq2 4406 . . . . . . . 8 ( = 𝐴 → {𝑣, } = {𝑣, 𝐴})
76eleq1d 2835 . . . . . . 7 ( = 𝐴 → ({𝑣, } ∈ 𝐸 ↔ {𝑣, 𝐴} ∈ 𝐸))
8 reueq1 3289 . . . . . . . 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 616 . . . . . 6 ( = 𝐴 → (({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {}){𝑣, 𝑤} ∈ 𝐸) ↔ ({𝑣, 𝐴} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {𝐴}){𝑣, 𝑤} ∈ 𝐸)))
115, 10raleqbidv 3301 . . . . 5 ( = 𝐴 → (∀𝑣 ∈ ({𝐴} ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {}){𝑣, 𝑤} ∈ 𝐸) ↔ ∀𝑣 ∈ ∅ ({𝑣, 𝐴} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {𝐴}){𝑣, 𝑤} ∈ 𝐸)))
1211rexsng 4358 . . . 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 466 . 2 ((𝐴𝑋𝑉 = {𝐴}) → ∃ ∈ {𝐴}∀𝑣 ∈ ({𝐴} ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {}){𝑣, 𝑤} ∈ 𝐸))
15 difeq1 3872 . . . . 5 (𝑉 = {𝐴} → (𝑉 ∖ {}) = ({𝐴} ∖ {}))
16 reueq1 3289 . . . . . . 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 614 . . . . 5 (𝑉 = {𝐴} → (({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸) ↔ ({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {}){𝑣, 𝑤} ∈ 𝐸)))
1915, 18raleqbidv 3301 . . . 4 (𝑉 = {𝐴} → (∀𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸) ↔ ∀𝑣 ∈ ({𝐴} ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {}){𝑣, 𝑤} ∈ 𝐸)))
2019rexeqbi1dv 3296 . . 3 (𝑉 = {𝐴} → (∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸) ↔ ∃ ∈ {𝐴}∀𝑣 ∈ ({𝐴} ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {}){𝑣, 𝑤} ∈ 𝐸)))
2120adantl 467 . 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 382   = wceq 1631   ∈ wcel 2145  ∀wral 3061  ∃wrex 3062  ∃!wreu 3063   ∖ cdif 3720  ∅c0 4063  {csn 4317  {cpr 4319 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 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-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-reu 3068  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-sn 4318  df-pr 4320 This theorem is referenced by:  1to2vfriswmgr  27461
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