Theorem cusgra2v 25991

Description: A graph with two (different) vertices is complete if and only if there is an edge between these two vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Proof shortened by Alexander van der Vekens, 16-Dec-2017.)

Assertion

Ref	Expression
cusgra2v	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({𝐴, 𝐵} USGrph 𝐸 → ({𝐴, 𝐵} ComplUSGrph 𝐸 ↔ {𝐴, 𝐵} ∈ ran 𝐸)))

Proof of Theorem cusgra2v

Dummy variables 𝑘 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		usgrav 25867	. . . . 5 ⊢ ({𝐴, 𝐵} USGrph 𝐸 → ({𝐴, 𝐵} ∈ V ∧ 𝐸 ∈ V))
2	1	adantr 480	. . . 4 ⊢ (({𝐴, 𝐵} USGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵)) → ({𝐴, 𝐵} ∈ V ∧ 𝐸 ∈ V))
3		iscusgra 25985	. . . 4 ⊢ (({𝐴, 𝐵} ∈ V ∧ 𝐸 ∈ V) → ({𝐴, 𝐵} ComplUSGrph 𝐸 ↔ ({𝐴, 𝐵} USGrph 𝐸 ∧ ∀𝑘 ∈ {𝐴, 𝐵}∀𝑛 ∈ ({𝐴, 𝐵} ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸)))
4	2, 3	syl 17	. . 3 ⊢ (({𝐴, 𝐵} USGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵)) → ({𝐴, 𝐵} ComplUSGrph 𝐸 ↔ ({𝐴, 𝐵} USGrph 𝐸 ∧ ∀𝑘 ∈ {𝐴, 𝐵}∀𝑛 ∈ ({𝐴, 𝐵} ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸)))
5		3simpa 1051	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊))
6	5	adantl 481	. . . . 5 ⊢ (({𝐴, 𝐵} USGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵)) → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊))
7		sneq 4135	. . . . . . . 8 ⊢ (𝑘 = 𝐴 → {𝑘} = {𝐴})
8	7	difeq2d 3690	. . . . . . 7 ⊢ (𝑘 = 𝐴 → ({𝐴, 𝐵} ∖ {𝑘}) = ({𝐴, 𝐵} ∖ {𝐴}))
9		preq2 4213	. . . . . . . 8 ⊢ (𝑘 = 𝐴 → {𝑛, 𝑘} = {𝑛, 𝐴})
10	9	eleq1d 2672	. . . . . . 7 ⊢ (𝑘 = 𝐴 → ({𝑛, 𝑘} ∈ ran 𝐸 ↔ {𝑛, 𝐴} ∈ ran 𝐸))
11	8, 10	raleqbidv 3129	. . . . . 6 ⊢ (𝑘 = 𝐴 → (∀𝑛 ∈ ({𝐴, 𝐵} ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸 ↔ ∀𝑛 ∈ ({𝐴, 𝐵} ∖ {𝐴}){𝑛, 𝐴} ∈ ran 𝐸))
12		sneq 4135	. . . . . . . 8 ⊢ (𝑘 = 𝐵 → {𝑘} = {𝐵})
13	12	difeq2d 3690	. . . . . . 7 ⊢ (𝑘 = 𝐵 → ({𝐴, 𝐵} ∖ {𝑘}) = ({𝐴, 𝐵} ∖ {𝐵}))
14		preq2 4213	. . . . . . . 8 ⊢ (𝑘 = 𝐵 → {𝑛, 𝑘} = {𝑛, 𝐵})
15	14	eleq1d 2672	. . . . . . 7 ⊢ (𝑘 = 𝐵 → ({𝑛, 𝑘} ∈ ran 𝐸 ↔ {𝑛, 𝐵} ∈ ran 𝐸))
16	13, 15	raleqbidv 3129	. . . . . 6 ⊢ (𝑘 = 𝐵 → (∀𝑛 ∈ ({𝐴, 𝐵} ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸 ↔ ∀𝑛 ∈ ({𝐴, 𝐵} ∖ {𝐵}){𝑛, 𝐵} ∈ ran 𝐸))
17	11, 16	ralprg 4181	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑘 ∈ {𝐴, 𝐵}∀𝑛 ∈ ({𝐴, 𝐵} ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸 ↔ (∀𝑛 ∈ ({𝐴, 𝐵} ∖ {𝐴}){𝑛, 𝐴} ∈ ran 𝐸 ∧ ∀𝑛 ∈ ({𝐴, 𝐵} ∖ {𝐵}){𝑛, 𝐵} ∈ ran 𝐸)))
18	6, 17	syl 17	. . . 4 ⊢ (({𝐴, 𝐵} USGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵)) → (∀𝑘 ∈ {𝐴, 𝐵}∀𝑛 ∈ ({𝐴, 𝐵} ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸 ↔ (∀𝑛 ∈ ({𝐴, 𝐵} ∖ {𝐴}){𝑛, 𝐴} ∈ ran 𝐸 ∧ ∀𝑛 ∈ ({𝐴, 𝐵} ∖ {𝐵}){𝑛, 𝐵} ∈ ran 𝐸)))
19		ibar 524	. . . . 5 ⊢ ({𝐴, 𝐵} USGrph 𝐸 → (∀𝑘 ∈ {𝐴, 𝐵}∀𝑛 ∈ ({𝐴, 𝐵} ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸 ↔ ({𝐴, 𝐵} USGrph 𝐸 ∧ ∀𝑘 ∈ {𝐴, 𝐵}∀𝑛 ∈ ({𝐴, 𝐵} ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸)))
20	19	adantr 480	. . . 4 ⊢ (({𝐴, 𝐵} USGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵)) → (∀𝑘 ∈ {𝐴, 𝐵}∀𝑛 ∈ ({𝐴, 𝐵} ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸 ↔ ({𝐴, 𝐵} USGrph 𝐸 ∧ ∀𝑘 ∈ {𝐴, 𝐵}∀𝑛 ∈ ({𝐴, 𝐵} ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸)))
21		difprsn1 4271	. . . . . . . . . 10 ⊢ (𝐴 ≠ 𝐵 → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵})
22	21	3ad2ant3 1077	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵})
23	22	adantl 481	. . . . . . . 8 ⊢ (({𝐴, 𝐵} USGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵)) → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵})
24	23	raleqdv 3121	. . . . . . 7 ⊢ (({𝐴, 𝐵} USGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵)) → (∀𝑛 ∈ ({𝐴, 𝐵} ∖ {𝐴}){𝑛, 𝐴} ∈ ran 𝐸 ↔ ∀𝑛 ∈ {𝐵} {𝑛, 𝐴} ∈ ran 𝐸))
25		preq1 4212	. . . . . . . . . . 11 ⊢ (𝑛 = 𝐵 → {𝑛, 𝐴} = {𝐵, 𝐴})
26	25	eleq1d 2672	. . . . . . . . . 10 ⊢ (𝑛 = 𝐵 → ({𝑛, 𝐴} ∈ ran 𝐸 ↔ {𝐵, 𝐴} ∈ ran 𝐸))
27	26	ralsng 4165	. . . . . . . . 9 ⊢ (𝐵 ∈ 𝑊 → (∀𝑛 ∈ {𝐵} {𝑛, 𝐴} ∈ ran 𝐸 ↔ {𝐵, 𝐴} ∈ ran 𝐸))
28	27	3ad2ant2 1076	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → (∀𝑛 ∈ {𝐵} {𝑛, 𝐴} ∈ ran 𝐸 ↔ {𝐵, 𝐴} ∈ ran 𝐸))
29	28	adantl 481	. . . . . . 7 ⊢ (({𝐴, 𝐵} USGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵)) → (∀𝑛 ∈ {𝐵} {𝑛, 𝐴} ∈ ran 𝐸 ↔ {𝐵, 𝐴} ∈ ran 𝐸))
30	24, 29	bitrd 267	. . . . . 6 ⊢ (({𝐴, 𝐵} USGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵)) → (∀𝑛 ∈ ({𝐴, 𝐵} ∖ {𝐴}){𝑛, 𝐴} ∈ ran 𝐸 ↔ {𝐵, 𝐴} ∈ ran 𝐸))
31		difprsn2 4272	. . . . . . . . . 10 ⊢ (𝐴 ≠ 𝐵 → ({𝐴, 𝐵} ∖ {𝐵}) = {𝐴})
32	31	3ad2ant3 1077	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({𝐴, 𝐵} ∖ {𝐵}) = {𝐴})
33	32	adantl 481	. . . . . . . 8 ⊢ (({𝐴, 𝐵} USGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵)) → ({𝐴, 𝐵} ∖ {𝐵}) = {𝐴})
34	33	raleqdv 3121	. . . . . . 7 ⊢ (({𝐴, 𝐵} USGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵)) → (∀𝑛 ∈ ({𝐴, 𝐵} ∖ {𝐵}){𝑛, 𝐵} ∈ ran 𝐸 ↔ ∀𝑛 ∈ {𝐴} {𝑛, 𝐵} ∈ ran 𝐸))
35		preq1 4212	. . . . . . . . . . 11 ⊢ (𝑛 = 𝐴 → {𝑛, 𝐵} = {𝐴, 𝐵})
36	35	eleq1d 2672	. . . . . . . . . 10 ⊢ (𝑛 = 𝐴 → ({𝑛, 𝐵} ∈ ran 𝐸 ↔ {𝐴, 𝐵} ∈ ran 𝐸))
37	36	ralsng 4165	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → (∀𝑛 ∈ {𝐴} {𝑛, 𝐵} ∈ ran 𝐸 ↔ {𝐴, 𝐵} ∈ ran 𝐸))
38	37	3ad2ant1 1075	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → (∀𝑛 ∈ {𝐴} {𝑛, 𝐵} ∈ ran 𝐸 ↔ {𝐴, 𝐵} ∈ ran 𝐸))
39	38	adantl 481	. . . . . . 7 ⊢ (({𝐴, 𝐵} USGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵)) → (∀𝑛 ∈ {𝐴} {𝑛, 𝐵} ∈ ran 𝐸 ↔ {𝐴, 𝐵} ∈ ran 𝐸))
40	34, 39	bitrd 267	. . . . . 6 ⊢ (({𝐴, 𝐵} USGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵)) → (∀𝑛 ∈ ({𝐴, 𝐵} ∖ {𝐵}){𝑛, 𝐵} ∈ ran 𝐸 ↔ {𝐴, 𝐵} ∈ ran 𝐸))
41	30, 40	anbi12d 743	. . . . 5 ⊢ (({𝐴, 𝐵} USGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵)) → ((∀𝑛 ∈ ({𝐴, 𝐵} ∖ {𝐴}){𝑛, 𝐴} ∈ ran 𝐸 ∧ ∀𝑛 ∈ ({𝐴, 𝐵} ∖ {𝐵}){𝑛, 𝐵} ∈ ran 𝐸) ↔ ({𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸)))
42		prcom 4211	. . . . . . . 8 ⊢ {𝐵, 𝐴} = {𝐴, 𝐵}
43	42	eleq1i 2679	. . . . . . 7 ⊢ ({𝐵, 𝐴} ∈ ran 𝐸 ↔ {𝐴, 𝐵} ∈ ran 𝐸)
44	43	anbi1i 727	. . . . . 6 ⊢ (({𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸) ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸))
45		anidm 674	. . . . . 6 ⊢ (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸) ↔ {𝐴, 𝐵} ∈ ran 𝐸)
46	44, 45	bitri 263	. . . . 5 ⊢ (({𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸) ↔ {𝐴, 𝐵} ∈ ran 𝐸)
47	41, 46	syl6bb 275	. . . 4 ⊢ (({𝐴, 𝐵} USGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵)) → ((∀𝑛 ∈ ({𝐴, 𝐵} ∖ {𝐴}){𝑛, 𝐴} ∈ ran 𝐸 ∧ ∀𝑛 ∈ ({𝐴, 𝐵} ∖ {𝐵}){𝑛, 𝐵} ∈ ran 𝐸) ↔ {𝐴, 𝐵} ∈ ran 𝐸))
48	18, 20, 47	3bitr3d 297	. . 3 ⊢ (({𝐴, 𝐵} USGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵)) → (({𝐴, 𝐵} USGrph 𝐸 ∧ ∀𝑘 ∈ {𝐴, 𝐵}∀𝑛 ∈ ({𝐴, 𝐵} ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸) ↔ {𝐴, 𝐵} ∈ ran 𝐸))
49	4, 48	bitrd 267	. 2 ⊢ (({𝐴, 𝐵} USGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵)) → ({𝐴, 𝐵} ComplUSGrph 𝐸 ↔ {𝐴, 𝐵} ∈ ran 𝐸))
50	49	expcom 450	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({𝐴, 𝐵} USGrph 𝐸 → ({𝐴, 𝐵} ComplUSGrph 𝐸 ↔ {𝐴, 𝐵} ∈ ran 𝐸)))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  Vcvv 3173   ∖ cdif 3537  {csn 4125  {cpr 4127   class class class wbr 4583  ran crn 5039   USGrph cusg 25859   ComplUSGrph ccusgra 25947

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-dm 5048  df-rn 5049  df-usgra 25862  df-cusgra 25950

This theorem is referenced by: (None)