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Theorem cusgra3v 25993
 Description: A graph with three (different) vertices is complete if and only if there is an edge between each of these three vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
Hypothesis
Ref Expression
cusgra3v.v 𝑉 = {𝐴, 𝐵, 𝐶}
Assertion
Ref Expression
cusgra3v (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝑉 USGrph 𝐸 ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (𝑉 ComplUSGrph 𝐸 ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)))

Proof of Theorem cusgra3v
Dummy variables 𝑘 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp2 1055 . . 3 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝑉 USGrph 𝐸 ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → 𝑉 USGrph 𝐸)
21biantrurd 528 . 2 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝑉 USGrph 𝐸 ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (∀𝑘𝑉𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸 ↔ (𝑉 USGrph 𝐸 ∧ ∀𝑘𝑉𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸)))
3 cusgra3v.v . . . . . . . 8 𝑉 = {𝐴, 𝐵, 𝐶}
43difeq1i 3686 . . . . . . 7 (𝑉 ∖ {𝑘}) = ({𝐴, 𝐵, 𝐶} ∖ {𝑘})
54a1i 11 . . . . . 6 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝑉 USGrph 𝐸 ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (𝑉 ∖ {𝑘}) = ({𝐴, 𝐵, 𝐶} ∖ {𝑘}))
65raleqdv 3121 . . . . 5 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝑉 USGrph 𝐸 ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (∀𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸 ↔ ∀𝑛 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸))
76ralbidv 2969 . . . 4 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝑉 USGrph 𝐸 ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (∀𝑘𝑉𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸 ↔ ∀𝑘𝑉𝑛 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸))
83raleqi 3119 . . . . 5 (∀𝑘𝑉𝑛 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸 ↔ ∀𝑘 ∈ {𝐴, 𝐵, 𝐶}∀𝑛 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸)
98a1i 11 . . . 4 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝑉 USGrph 𝐸 ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (∀𝑘𝑉𝑛 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸 ↔ ∀𝑘 ∈ {𝐴, 𝐵, 𝐶}∀𝑛 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸))
10 sneq 4135 . . . . . . . 8 (𝑘 = 𝐴 → {𝑘} = {𝐴})
1110difeq2d 3690 . . . . . . 7 (𝑘 = 𝐴 → ({𝐴, 𝐵, 𝐶} ∖ {𝑘}) = ({𝐴, 𝐵, 𝐶} ∖ {𝐴}))
12 preq2 4213 . . . . . . . 8 (𝑘 = 𝐴 → {𝑛, 𝑘} = {𝑛, 𝐴})
1312eleq1d 2672 . . . . . . 7 (𝑘 = 𝐴 → ({𝑛, 𝑘} ∈ ran 𝐸 ↔ {𝑛, 𝐴} ∈ ran 𝐸))
1411, 13raleqbidv 3129 . . . . . 6 (𝑘 = 𝐴 → (∀𝑛 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸 ↔ ∀𝑛 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴}){𝑛, 𝐴} ∈ ran 𝐸))
15 sneq 4135 . . . . . . . 8 (𝑘 = 𝐵 → {𝑘} = {𝐵})
1615difeq2d 3690 . . . . . . 7 (𝑘 = 𝐵 → ({𝐴, 𝐵, 𝐶} ∖ {𝑘}) = ({𝐴, 𝐵, 𝐶} ∖ {𝐵}))
17 preq2 4213 . . . . . . . 8 (𝑘 = 𝐵 → {𝑛, 𝑘} = {𝑛, 𝐵})
1817eleq1d 2672 . . . . . . 7 (𝑘 = 𝐵 → ({𝑛, 𝑘} ∈ ran 𝐸 ↔ {𝑛, 𝐵} ∈ ran 𝐸))
1916, 18raleqbidv 3129 . . . . . 6 (𝑘 = 𝐵 → (∀𝑛 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸 ↔ ∀𝑛 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵}){𝑛, 𝐵} ∈ ran 𝐸))
20 sneq 4135 . . . . . . . 8 (𝑘 = 𝐶 → {𝑘} = {𝐶})
2120difeq2d 3690 . . . . . . 7 (𝑘 = 𝐶 → ({𝐴, 𝐵, 𝐶} ∖ {𝑘}) = ({𝐴, 𝐵, 𝐶} ∖ {𝐶}))
22 preq2 4213 . . . . . . . 8 (𝑘 = 𝐶 → {𝑛, 𝑘} = {𝑛, 𝐶})
2322eleq1d 2672 . . . . . . 7 (𝑘 = 𝐶 → ({𝑛, 𝑘} ∈ ran 𝐸 ↔ {𝑛, 𝐶} ∈ ran 𝐸))
2421, 23raleqbidv 3129 . . . . . 6 (𝑘 = 𝐶 → (∀𝑛 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸 ↔ ∀𝑛 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑛, 𝐶} ∈ ran 𝐸))
2514, 19, 24raltpg 4183 . . . . 5 ((𝐴𝑋𝐵𝑌𝐶𝑍) → (∀𝑘 ∈ {𝐴, 𝐵, 𝐶}∀𝑛 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸 ↔ (∀𝑛 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴}){𝑛, 𝐴} ∈ ran 𝐸 ∧ ∀𝑛 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵}){𝑛, 𝐵} ∈ ran 𝐸 ∧ ∀𝑛 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑛, 𝐶} ∈ ran 𝐸)))
26253ad2ant1 1075 . . . 4 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝑉 USGrph 𝐸 ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (∀𝑘 ∈ {𝐴, 𝐵, 𝐶}∀𝑛 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸 ↔ (∀𝑛 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴}){𝑛, 𝐴} ∈ ran 𝐸 ∧ ∀𝑛 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵}){𝑛, 𝐵} ∈ ran 𝐸 ∧ ∀𝑛 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑛, 𝐶} ∈ ran 𝐸)))
277, 9, 263bitrd 293 . . 3 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝑉 USGrph 𝐸 ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (∀𝑘𝑉𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸 ↔ (∀𝑛 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴}){𝑛, 𝐴} ∈ ran 𝐸 ∧ ∀𝑛 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵}){𝑛, 𝐵} ∈ ran 𝐸 ∧ ∀𝑛 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑛, 𝐶} ∈ ran 𝐸)))
28 tprot 4228 . . . . . . . . 9 {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴}
2928a1i 11 . . . . . . . 8 ((𝐴𝐵𝐴𝐶𝐵𝐶) → {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴})
3029difeq1d 3689 . . . . . . 7 ((𝐴𝐵𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐴}) = ({𝐵, 𝐶, 𝐴} ∖ {𝐴}))
31 necom 2835 . . . . . . . . . . 11 (𝐴𝐵𝐵𝐴)
32 necom 2835 . . . . . . . . . . 11 (𝐴𝐶𝐶𝐴)
3331, 32anbi12i 729 . . . . . . . . . 10 ((𝐴𝐵𝐴𝐶) ↔ (𝐵𝐴𝐶𝐴))
3433biimpi 205 . . . . . . . . 9 ((𝐴𝐵𝐴𝐶) → (𝐵𝐴𝐶𝐴))
35343adant3 1074 . . . . . . . 8 ((𝐴𝐵𝐴𝐶𝐵𝐶) → (𝐵𝐴𝐶𝐴))
36 diftpsn3 4273 . . . . . . . 8 ((𝐵𝐴𝐶𝐴) → ({𝐵, 𝐶, 𝐴} ∖ {𝐴}) = {𝐵, 𝐶})
3735, 36syl 17 . . . . . . 7 ((𝐴𝐵𝐴𝐶𝐵𝐶) → ({𝐵, 𝐶, 𝐴} ∖ {𝐴}) = {𝐵, 𝐶})
3830, 37eqtrd 2644 . . . . . 6 ((𝐴𝐵𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐴}) = {𝐵, 𝐶})
39383ad2ant3 1077 . . . . 5 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝑉 USGrph 𝐸 ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ({𝐴, 𝐵, 𝐶} ∖ {𝐴}) = {𝐵, 𝐶})
4039raleqdv 3121 . . . 4 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝑉 USGrph 𝐸 ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (∀𝑛 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴}){𝑛, 𝐴} ∈ ran 𝐸 ↔ ∀𝑛 ∈ {𝐵, 𝐶} {𝑛, 𝐴} ∈ ran 𝐸))
41 tpcomb 4230 . . . . . . . . 9 {𝐴, 𝐵, 𝐶} = {𝐴, 𝐶, 𝐵}
4241a1i 11 . . . . . . . 8 ((𝐴𝐵𝐴𝐶𝐵𝐶) → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐶, 𝐵})
4342difeq1d 3689 . . . . . . 7 ((𝐴𝐵𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐵}) = ({𝐴, 𝐶, 𝐵} ∖ {𝐵}))
44 necom 2835 . . . . . . . . . . 11 (𝐵𝐶𝐶𝐵)
4544biimpi 205 . . . . . . . . . 10 (𝐵𝐶𝐶𝐵)
4645anim2i 591 . . . . . . . . 9 ((𝐴𝐵𝐵𝐶) → (𝐴𝐵𝐶𝐵))
47463adant2 1073 . . . . . . . 8 ((𝐴𝐵𝐴𝐶𝐵𝐶) → (𝐴𝐵𝐶𝐵))
48 diftpsn3 4273 . . . . . . . 8 ((𝐴𝐵𝐶𝐵) → ({𝐴, 𝐶, 𝐵} ∖ {𝐵}) = {𝐴, 𝐶})
4947, 48syl 17 . . . . . . 7 ((𝐴𝐵𝐴𝐶𝐵𝐶) → ({𝐴, 𝐶, 𝐵} ∖ {𝐵}) = {𝐴, 𝐶})
5043, 49eqtrd 2644 . . . . . 6 ((𝐴𝐵𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐵}) = {𝐴, 𝐶})
51503ad2ant3 1077 . . . . 5 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝑉 USGrph 𝐸 ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ({𝐴, 𝐵, 𝐶} ∖ {𝐵}) = {𝐴, 𝐶})
5251raleqdv 3121 . . . 4 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝑉 USGrph 𝐸 ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (∀𝑛 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵}){𝑛, 𝐵} ∈ ran 𝐸 ↔ ∀𝑛 ∈ {𝐴, 𝐶} {𝑛, 𝐵} ∈ ran 𝐸))
53 diftpsn3 4273 . . . . . . 7 ((𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐶}) = {𝐴, 𝐵})
54533adant1 1072 . . . . . 6 ((𝐴𝐵𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐶}) = {𝐴, 𝐵})
55543ad2ant3 1077 . . . . 5 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝑉 USGrph 𝐸 ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ({𝐴, 𝐵, 𝐶} ∖ {𝐶}) = {𝐴, 𝐵})
5655raleqdv 3121 . . . 4 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝑉 USGrph 𝐸 ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (∀𝑛 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑛, 𝐶} ∈ ran 𝐸 ↔ ∀𝑛 ∈ {𝐴, 𝐵} {𝑛, 𝐶} ∈ ran 𝐸))
5740, 52, 563anbi123d 1391 . . 3 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝑉 USGrph 𝐸 ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ((∀𝑛 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴}){𝑛, 𝐴} ∈ ran 𝐸 ∧ ∀𝑛 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵}){𝑛, 𝐵} ∈ ran 𝐸 ∧ ∀𝑛 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑛, 𝐶} ∈ ran 𝐸) ↔ (∀𝑛 ∈ {𝐵, 𝐶} {𝑛, 𝐴} ∈ ran 𝐸 ∧ ∀𝑛 ∈ {𝐴, 𝐶} {𝑛, 𝐵} ∈ ran 𝐸 ∧ ∀𝑛 ∈ {𝐴, 𝐵} {𝑛, 𝐶} ∈ ran 𝐸)))
58 preq1 4212 . . . . . . . . 9 (𝑛 = 𝐵 → {𝑛, 𝐴} = {𝐵, 𝐴})
5958eleq1d 2672 . . . . . . . 8 (𝑛 = 𝐵 → ({𝑛, 𝐴} ∈ ran 𝐸 ↔ {𝐵, 𝐴} ∈ ran 𝐸))
60 preq1 4212 . . . . . . . . 9 (𝑛 = 𝐶 → {𝑛, 𝐴} = {𝐶, 𝐴})
6160eleq1d 2672 . . . . . . . 8 (𝑛 = 𝐶 → ({𝑛, 𝐴} ∈ ran 𝐸 ↔ {𝐶, 𝐴} ∈ ran 𝐸))
6259, 61ralprg 4181 . . . . . . 7 ((𝐵𝑌𝐶𝑍) → (∀𝑛 ∈ {𝐵, 𝐶} {𝑛, 𝐴} ∈ ran 𝐸 ↔ ({𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)))
63623adant1 1072 . . . . . 6 ((𝐴𝑋𝐵𝑌𝐶𝑍) → (∀𝑛 ∈ {𝐵, 𝐶} {𝑛, 𝐴} ∈ ran 𝐸 ↔ ({𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)))
64633ad2ant1 1075 . . . . 5 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝑉 USGrph 𝐸 ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (∀𝑛 ∈ {𝐵, 𝐶} {𝑛, 𝐴} ∈ ran 𝐸 ↔ ({𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)))
65 preq1 4212 . . . . . . . . 9 (𝑛 = 𝐴 → {𝑛, 𝐵} = {𝐴, 𝐵})
6665eleq1d 2672 . . . . . . . 8 (𝑛 = 𝐴 → ({𝑛, 𝐵} ∈ ran 𝐸 ↔ {𝐴, 𝐵} ∈ ran 𝐸))
67 preq1 4212 . . . . . . . . 9 (𝑛 = 𝐶 → {𝑛, 𝐵} = {𝐶, 𝐵})
6867eleq1d 2672 . . . . . . . 8 (𝑛 = 𝐶 → ({𝑛, 𝐵} ∈ ran 𝐸 ↔ {𝐶, 𝐵} ∈ ran 𝐸))
6966, 68ralprg 4181 . . . . . . 7 ((𝐴𝑋𝐶𝑍) → (∀𝑛 ∈ {𝐴, 𝐶} {𝑛, 𝐵} ∈ ran 𝐸 ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸)))
70693adant2 1073 . . . . . 6 ((𝐴𝑋𝐵𝑌𝐶𝑍) → (∀𝑛 ∈ {𝐴, 𝐶} {𝑛, 𝐵} ∈ ran 𝐸 ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸)))
71703ad2ant1 1075 . . . . 5 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝑉 USGrph 𝐸 ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (∀𝑛 ∈ {𝐴, 𝐶} {𝑛, 𝐵} ∈ ran 𝐸 ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸)))
72 preq1 4212 . . . . . . . . 9 (𝑛 = 𝐴 → {𝑛, 𝐶} = {𝐴, 𝐶})
7372eleq1d 2672 . . . . . . . 8 (𝑛 = 𝐴 → ({𝑛, 𝐶} ∈ ran 𝐸 ↔ {𝐴, 𝐶} ∈ ran 𝐸))
74 preq1 4212 . . . . . . . . 9 (𝑛 = 𝐵 → {𝑛, 𝐶} = {𝐵, 𝐶})
7574eleq1d 2672 . . . . . . . 8 (𝑛 = 𝐵 → ({𝑛, 𝐶} ∈ ran 𝐸 ↔ {𝐵, 𝐶} ∈ ran 𝐸))
7673, 75ralprg 4181 . . . . . . 7 ((𝐴𝑋𝐵𝑌) → (∀𝑛 ∈ {𝐴, 𝐵} {𝑛, 𝐶} ∈ ran 𝐸 ↔ ({𝐴, 𝐶} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)))
77763adant3 1074 . . . . . 6 ((𝐴𝑋𝐵𝑌𝐶𝑍) → (∀𝑛 ∈ {𝐴, 𝐵} {𝑛, 𝐶} ∈ ran 𝐸 ↔ ({𝐴, 𝐶} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)))
78773ad2ant1 1075 . . . . 5 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝑉 USGrph 𝐸 ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (∀𝑛 ∈ {𝐴, 𝐵} {𝑛, 𝐶} ∈ ran 𝐸 ↔ ({𝐴, 𝐶} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)))
7964, 71, 783anbi123d 1391 . . . 4 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝑉 USGrph 𝐸 ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ((∀𝑛 ∈ {𝐵, 𝐶} {𝑛, 𝐴} ∈ ran 𝐸 ∧ ∀𝑛 ∈ {𝐴, 𝐶} {𝑛, 𝐵} ∈ ran 𝐸 ∧ ∀𝑛 ∈ {𝐴, 𝐵} {𝑛, 𝐶} ∈ ran 𝐸) ↔ (({𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸) ∧ ({𝐴, 𝐶} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸))))
80 ancom 465 . . . . . . 7 (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸) ↔ ({𝐶, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸))
81803anbi2i 1247 . . . . . 6 ((({𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸) ∧ ({𝐴, 𝐶} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) ↔ (({𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) ∧ ({𝐶, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸) ∧ ({𝐴, 𝐶} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)))
82 3an6 1401 . . . . . 6 ((({𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) ∧ ({𝐶, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸) ∧ ({𝐴, 𝐶} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) ↔ (({𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸) ∧ ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)))
8381, 82bitri 263 . . . . 5 ((({𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸) ∧ ({𝐴, 𝐶} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) ↔ (({𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸) ∧ ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)))
8483a1i 11 . . . 4 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝑉 USGrph 𝐸 ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ((({𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸) ∧ ({𝐴, 𝐶} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) ↔ (({𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸) ∧ ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸))))
85 prcom 4211 . . . . . . . . 9 {𝐵, 𝐴} = {𝐴, 𝐵}
8685eleq1i 2679 . . . . . . . 8 ({𝐵, 𝐴} ∈ ran 𝐸 ↔ {𝐴, 𝐵} ∈ ran 𝐸)
87 prcom 4211 . . . . . . . . 9 {𝐶, 𝐵} = {𝐵, 𝐶}
8887eleq1i 2679 . . . . . . . 8 ({𝐶, 𝐵} ∈ ran 𝐸 ↔ {𝐵, 𝐶} ∈ ran 𝐸)
89 prcom 4211 . . . . . . . . 9 {𝐴, 𝐶} = {𝐶, 𝐴}
9089eleq1i 2679 . . . . . . . 8 ({𝐴, 𝐶} ∈ ran 𝐸 ↔ {𝐶, 𝐴} ∈ ran 𝐸)
9186, 88, 903anbi123i 1244 . . . . . . 7 (({𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸) ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))
92 3anrot 1036 . . . . . . 7 (({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))
9391, 92anbi12i 729 . . . . . 6 ((({𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸) ∧ ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) ↔ (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)))
94 anidm 674 . . . . . 6 ((({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))
9593, 94bitri 263 . . . . 5 ((({𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸) ∧ ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))
9695a1i 11 . . . 4 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝑉 USGrph 𝐸 ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ((({𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸) ∧ ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)))
9779, 84, 963bitrd 293 . . 3 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝑉 USGrph 𝐸 ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ((∀𝑛 ∈ {𝐵, 𝐶} {𝑛, 𝐴} ∈ ran 𝐸 ∧ ∀𝑛 ∈ {𝐴, 𝐶} {𝑛, 𝐵} ∈ ran 𝐸 ∧ ∀𝑛 ∈ {𝐴, 𝐵} {𝑛, 𝐶} ∈ ran 𝐸) ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)))
9827, 57, 973bitrrd 294 . 2 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝑉 USGrph 𝐸 ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) ↔ ∀𝑘𝑉𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸))
99 usgrav 25867 . . . 4 (𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
100 iscusgra 25985 . . . 4 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉 ComplUSGrph 𝐸 ↔ (𝑉 USGrph 𝐸 ∧ ∀𝑘𝑉𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸)))
10199, 100syl 17 . . 3 (𝑉 USGrph 𝐸 → (𝑉 ComplUSGrph 𝐸 ↔ (𝑉 USGrph 𝐸 ∧ ∀𝑘𝑉𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸)))
1021013ad2ant2 1076 . 2 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝑉 USGrph 𝐸 ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (𝑉 ComplUSGrph 𝐸 ↔ (𝑉 USGrph 𝐸 ∧ ∀𝑘𝑉𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸)))
1032, 98, 1023bitr4rd 300 1 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝑉 USGrph 𝐸 ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (𝑉 ComplUSGrph 𝐸 ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)))
 Colors of variables: wff setvar class 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  {ctp 4129   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-3or 1032  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-tp 4130  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:  cusgra3vnbpr  25994
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