Theorem 1vwmgr 30810

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	⊢ ((𝐴 ∈ 𝑋 ∧ 𝑉 = {𝐴}) → ∃ℎ ∈ 𝑉 ∀𝑣 ∈ (𝑉 ∖ {ℎ})({𝑣, ℎ} ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸))

Distinct variable groups:   𝐴,ℎ,𝑣,𝑤   ℎ,𝐸   ℎ,𝑉,𝑣,𝑤

Allowed substitution hints:   𝐸(𝑤, 𝑣)   𝑋(𝑤, 𝑣, ℎ)

Proof of Theorem 1vwmgr

Step	Hyp	Ref	Expression
1		ral0 4453	. . . 4 ⊢ ∀𝑣 ∈ ∅ ({𝑣, 𝐴} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {𝐴}){𝑣, 𝑤} ∈ 𝐸)
2		sneq 4593	. . . . . . . 8 ⊢ (ℎ = 𝐴 → {ℎ} = {𝐴})
3	2	difeq2d 4073	. . . . . . 7 ⊢ (ℎ = 𝐴 → ({𝐴} ∖ {ℎ}) = ({𝐴} ∖ {𝐴}))
4		difid 4324	. . . . . . 7 ⊢ ({𝐴} ∖ {𝐴}) = ∅
5	3, 4	eqtrdi 2811	. . . . . 6 ⊢ (ℎ = 𝐴 → ({𝐴} ∖ {ℎ}) = ∅)
6		preq2 4694	. . . . . . . 8 ⊢ (ℎ = 𝐴 → {𝑣, ℎ} = {𝑣, 𝐴})
7	6	eleq1d 2845	. . . . . . 7 ⊢ (ℎ = 𝐴 → ({𝑣, ℎ} ∈ 𝐸 ↔ {𝑣, 𝐴} ∈ 𝐸))
8		reueq1 3397	. . . . . . . 8 ⊢ (({𝐴} ∖ {ℎ}) = ({𝐴} ∖ {𝐴}) → (∃!𝑤 ∈ ({𝐴} ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸 ↔ ∃!𝑤 ∈ ({𝐴} ∖ {𝐴}){𝑣, 𝑤} ∈ 𝐸))
9	3, 8	syl 18	. . . . . . 7 ⊢ (ℎ = 𝐴 → (∃!𝑤 ∈ ({𝐴} ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸 ↔ ∃!𝑤 ∈ ({𝐴} ∖ {𝐴}){𝑣, 𝑤} ∈ 𝐸))
10	7, 9	anbi12d 644	. . . . . 6 ⊢ (ℎ = 𝐴 → (({𝑣, ℎ} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸) ↔ ({𝑣, 𝐴} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {𝐴}){𝑣, 𝑤} ∈ 𝐸)))
11	5, 10	raleqbidv 3334	. . . . 5 ⊢ (ℎ = 𝐴 → (∀𝑣 ∈ ({𝐴} ∖ {ℎ})({𝑣, ℎ} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸) ↔ ∀𝑣 ∈ ∅ ({𝑣, 𝐴} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {𝐴}){𝑣, 𝑤} ∈ 𝐸)))
12	11	rexsng 4636	. . . 4 ⊢ (𝐴 ∈ 𝑋 → (∃ℎ ∈ {𝐴}∀𝑣 ∈ ({𝐴} ∖ {ℎ})({𝑣, ℎ} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸) ↔ ∀𝑣 ∈ ∅ ({𝑣, 𝐴} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {𝐴}){𝑣, 𝑤} ∈ 𝐸)))
13	1, 12	mpbiri 261	. . 3 ⊢ (𝐴 ∈ 𝑋 → ∃ℎ ∈ {𝐴}∀𝑣 ∈ ({𝐴} ∖ {ℎ})({𝑣, ℎ} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸))
14	13	adantr 486	. 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝑉 = {𝐴}) → ∃ℎ ∈ {𝐴}∀𝑣 ∈ ({𝐴} ∖ {ℎ})({𝑣, ℎ} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸))
15		difeq1 4066	. . . . 5 ⊢ (𝑉 = {𝐴} → (𝑉 ∖ {ℎ}) = ({𝐴} ∖ {ℎ}))
16		reueq1 3397	. . . . . . 7 ⊢ ((𝑉 ∖ {ℎ}) = ({𝐴} ∖ {ℎ}) → (∃!𝑤 ∈ (𝑉 ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸 ↔ ∃!𝑤 ∈ ({𝐴} ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸))
17	15, 16	syl 18	. . . . . 6 ⊢ (𝑉 = {𝐴} → (∃!𝑤 ∈ (𝑉 ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸 ↔ ∃!𝑤 ∈ ({𝐴} ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸))
18	17	anbi2d 642	. . . . 5 ⊢ (𝑉 = {𝐴} → (({𝑣, ℎ} ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸) ↔ ({𝑣, ℎ} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸)))
19	15, 18	raleqbidv 3334	. . . 4 ⊢ (𝑉 = {𝐴} → (∀𝑣 ∈ (𝑉 ∖ {ℎ})({𝑣, ℎ} ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸) ↔ ∀𝑣 ∈ ({𝐴} ∖ {ℎ})({𝑣, ℎ} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸)))
20	19	rexeqbi1dv 3330	. . 3 ⊢ (𝑉 = {𝐴} → (∃ℎ ∈ 𝑉 ∀𝑣 ∈ (𝑉 ∖ {ℎ})({𝑣, ℎ} ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸) ↔ ∃ℎ ∈ {𝐴}∀𝑣 ∈ ({𝐴} ∖ {ℎ})({𝑣, ℎ} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸)))
21	20	adantl 487	. 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝑉 = {𝐴}) → (∃ℎ ∈ 𝑉 ∀𝑣 ∈ (𝑉 ∖ {ℎ})({𝑣, ℎ} ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸) ↔ ∃ℎ ∈ {𝐴}∀𝑣 ∈ ({𝐴} ∖ {ℎ})({𝑣, ℎ} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸)))
22	14, 21	mpbird 260	1 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝑉 = {𝐴}) → ∃ℎ ∈ 𝑉 ∀𝑣 ∈ (𝑉 ∖ {ℎ})({𝑣, ℎ} ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076  ∃wrex 3086  ∃!wreu 3363   ∖ cdif 3895  ∅c0 4278  {csn 4583  {cpr 4585

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-nul 4279  df-sn 4584  df-pr 4586

This theorem is used by:  1to2vfriswmgr  30813