Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  nb3grprlem2 Structured version   Visualization version   GIF version

Theorem nb3grprlem2 40609
 Description: Lemma 2 for nb3grapr 25982. (Contributed by Alexander van der Vekens, 17-Oct-2017.) (Revised by AV, 28-Oct-2020.)
Hypotheses
Ref Expression
nb3grpr.v 𝑉 = (Vtx‘𝐺)
nb3grpr.e 𝐸 = (Edg‘𝐺)
nb3grpr.g (𝜑𝐺 ∈ USGraph )
nb3grpr.t (𝜑𝑉 = {𝐴, 𝐵, 𝐶})
nb3grpr.s (𝜑 → (𝐴𝑋𝐵𝑌𝐶𝑍))
nb3grpr.n (𝜑 → (𝐴𝐵𝐴𝐶𝐵𝐶))
Assertion
Ref Expression
nb3grprlem2 (𝜑 → ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ↔ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣})(𝐺 NeighbVtx 𝐴) = {𝑣, 𝑤}))
Distinct variable groups:   𝑣,𝐴   𝑣,𝐵   𝑣,𝐶   𝑣,𝐸   𝑣,𝐺   𝑣,𝑉   𝜑,𝑣   𝑤,𝐴,𝑣   𝑤,𝐵   𝑤,𝐶   𝑤,𝐺   𝑤,𝑉
Allowed substitution hints:   𝜑(𝑤)   𝐸(𝑤)   𝑋(𝑤,𝑣)   𝑌(𝑤,𝑣)   𝑍(𝑤,𝑣)

Proof of Theorem nb3grprlem2
StepHypRef Expression
1 nb3grpr.s . . 3 (𝜑 → (𝐴𝑋𝐵𝑌𝐶𝑍))
2 sneq 4135 . . . . . 6 (𝑣 = 𝐴 → {𝑣} = {𝐴})
32difeq2d 3690 . . . . 5 (𝑣 = 𝐴 → ({𝐴, 𝐵, 𝐶} ∖ {𝑣}) = ({𝐴, 𝐵, 𝐶} ∖ {𝐴}))
4 preq1 4212 . . . . . 6 (𝑣 = 𝐴 → {𝑣, 𝑤} = {𝐴, 𝑤})
54eqeq2d 2620 . . . . 5 (𝑣 = 𝐴 → ((𝐺 NeighbVtx 𝐴) = {𝑣, 𝑤} ↔ (𝐺 NeighbVtx 𝐴) = {𝐴, 𝑤}))
63, 5rexeqbidv 3130 . . . 4 (𝑣 = 𝐴 → (∃𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝑣})(𝐺 NeighbVtx 𝐴) = {𝑣, 𝑤} ↔ ∃𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴})(𝐺 NeighbVtx 𝐴) = {𝐴, 𝑤}))
7 sneq 4135 . . . . . 6 (𝑣 = 𝐵 → {𝑣} = {𝐵})
87difeq2d 3690 . . . . 5 (𝑣 = 𝐵 → ({𝐴, 𝐵, 𝐶} ∖ {𝑣}) = ({𝐴, 𝐵, 𝐶} ∖ {𝐵}))
9 preq1 4212 . . . . . 6 (𝑣 = 𝐵 → {𝑣, 𝑤} = {𝐵, 𝑤})
109eqeq2d 2620 . . . . 5 (𝑣 = 𝐵 → ((𝐺 NeighbVtx 𝐴) = {𝑣, 𝑤} ↔ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝑤}))
118, 10rexeqbidv 3130 . . . 4 (𝑣 = 𝐵 → (∃𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝑣})(𝐺 NeighbVtx 𝐴) = {𝑣, 𝑤} ↔ ∃𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵})(𝐺 NeighbVtx 𝐴) = {𝐵, 𝑤}))
12 sneq 4135 . . . . . 6 (𝑣 = 𝐶 → {𝑣} = {𝐶})
1312difeq2d 3690 . . . . 5 (𝑣 = 𝐶 → ({𝐴, 𝐵, 𝐶} ∖ {𝑣}) = ({𝐴, 𝐵, 𝐶} ∖ {𝐶}))
14 preq1 4212 . . . . . 6 (𝑣 = 𝐶 → {𝑣, 𝑤} = {𝐶, 𝑤})
1514eqeq2d 2620 . . . . 5 (𝑣 = 𝐶 → ((𝐺 NeighbVtx 𝐴) = {𝑣, 𝑤} ↔ (𝐺 NeighbVtx 𝐴) = {𝐶, 𝑤}))
1613, 15rexeqbidv 3130 . . . 4 (𝑣 = 𝐶 → (∃𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝑣})(𝐺 NeighbVtx 𝐴) = {𝑣, 𝑤} ↔ ∃𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶})(𝐺 NeighbVtx 𝐴) = {𝐶, 𝑤}))
176, 11, 16rextpg 4184 . . 3 ((𝐴𝑋𝐵𝑌𝐶𝑍) → (∃𝑣 ∈ {𝐴, 𝐵, 𝐶}∃𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝑣})(𝐺 NeighbVtx 𝐴) = {𝑣, 𝑤} ↔ (∃𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴})(𝐺 NeighbVtx 𝐴) = {𝐴, 𝑤} ∨ ∃𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵})(𝐺 NeighbVtx 𝐴) = {𝐵, 𝑤} ∨ ∃𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶})(𝐺 NeighbVtx 𝐴) = {𝐶, 𝑤})))
181, 17syl 17 . 2 (𝜑 → (∃𝑣 ∈ {𝐴, 𝐵, 𝐶}∃𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝑣})(𝐺 NeighbVtx 𝐴) = {𝑣, 𝑤} ↔ (∃𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴})(𝐺 NeighbVtx 𝐴) = {𝐴, 𝑤} ∨ ∃𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵})(𝐺 NeighbVtx 𝐴) = {𝐵, 𝑤} ∨ ∃𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶})(𝐺 NeighbVtx 𝐴) = {𝐶, 𝑤})))
19 nb3grpr.t . . . 4 (𝜑𝑉 = {𝐴, 𝐵, 𝐶})
20 nb3grpr.g . . . 4 (𝜑𝐺 ∈ USGraph )
2119, 20jca 553 . . 3 (𝜑 → (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph ))
22 simpl 472 . . . 4 ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph ) → 𝑉 = {𝐴, 𝐵, 𝐶})
23 difeq1 3683 . . . . . 6 (𝑉 = {𝐴, 𝐵, 𝐶} → (𝑉 ∖ {𝑣}) = ({𝐴, 𝐵, 𝐶} ∖ {𝑣}))
2423adantr 480 . . . . 5 ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph ) → (𝑉 ∖ {𝑣}) = ({𝐴, 𝐵, 𝐶} ∖ {𝑣}))
2524rexeqdv 3122 . . . 4 ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph ) → (∃𝑤 ∈ (𝑉 ∖ {𝑣})(𝐺 NeighbVtx 𝐴) = {𝑣, 𝑤} ↔ ∃𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝑣})(𝐺 NeighbVtx 𝐴) = {𝑣, 𝑤}))
2622, 25rexeqbidv 3130 . . 3 ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph ) → (∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣})(𝐺 NeighbVtx 𝐴) = {𝑣, 𝑤} ↔ ∃𝑣 ∈ {𝐴, 𝐵, 𝐶}∃𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝑣})(𝐺 NeighbVtx 𝐴) = {𝑣, 𝑤}))
2721, 26syl 17 . 2 (𝜑 → (∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣})(𝐺 NeighbVtx 𝐴) = {𝑣, 𝑤} ↔ ∃𝑣 ∈ {𝐴, 𝐵, 𝐶}∃𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝑣})(𝐺 NeighbVtx 𝐴) = {𝑣, 𝑤}))
28 preq2 4213 . . . . . . . 8 (𝑤 = 𝐵 → {𝐴, 𝑤} = {𝐴, 𝐵})
2928eqeq2d 2620 . . . . . . 7 (𝑤 = 𝐵 → ((𝐺 NeighbVtx 𝐴) = {𝐴, 𝑤} ↔ (𝐺 NeighbVtx 𝐴) = {𝐴, 𝐵}))
30 preq2 4213 . . . . . . . 8 (𝑤 = 𝐶 → {𝐴, 𝑤} = {𝐴, 𝐶})
3130eqeq2d 2620 . . . . . . 7 (𝑤 = 𝐶 → ((𝐺 NeighbVtx 𝐴) = {𝐴, 𝑤} ↔ (𝐺 NeighbVtx 𝐴) = {𝐴, 𝐶}))
3229, 31rexprg 4182 . . . . . 6 ((𝐵𝑌𝐶𝑍) → (∃𝑤 ∈ {𝐵, 𝐶} (𝐺 NeighbVtx 𝐴) = {𝐴, 𝑤} ↔ ((𝐺 NeighbVtx 𝐴) = {𝐴, 𝐵} ∨ (𝐺 NeighbVtx 𝐴) = {𝐴, 𝐶})))
33323adant1 1072 . . . . 5 ((𝐴𝑋𝐵𝑌𝐶𝑍) → (∃𝑤 ∈ {𝐵, 𝐶} (𝐺 NeighbVtx 𝐴) = {𝐴, 𝑤} ↔ ((𝐺 NeighbVtx 𝐴) = {𝐴, 𝐵} ∨ (𝐺 NeighbVtx 𝐴) = {𝐴, 𝐶})))
34 preq2 4213 . . . . . . . . 9 (𝑤 = 𝐶 → {𝐵, 𝑤} = {𝐵, 𝐶})
3534eqeq2d 2620 . . . . . . . 8 (𝑤 = 𝐶 → ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝑤} ↔ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}))
36 preq2 4213 . . . . . . . . 9 (𝑤 = 𝐴 → {𝐵, 𝑤} = {𝐵, 𝐴})
3736eqeq2d 2620 . . . . . . . 8 (𝑤 = 𝐴 → ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝑤} ↔ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐴}))
3835, 37rexprg 4182 . . . . . . 7 ((𝐶𝑍𝐴𝑋) → (∃𝑤 ∈ {𝐶, 𝐴} (𝐺 NeighbVtx 𝐴) = {𝐵, 𝑤} ↔ ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ∨ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐴})))
3938ancoms 468 . . . . . 6 ((𝐴𝑋𝐶𝑍) → (∃𝑤 ∈ {𝐶, 𝐴} (𝐺 NeighbVtx 𝐴) = {𝐵, 𝑤} ↔ ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ∨ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐴})))
40393adant2 1073 . . . . 5 ((𝐴𝑋𝐵𝑌𝐶𝑍) → (∃𝑤 ∈ {𝐶, 𝐴} (𝐺 NeighbVtx 𝐴) = {𝐵, 𝑤} ↔ ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ∨ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐴})))
41 preq2 4213 . . . . . . . 8 (𝑤 = 𝐴 → {𝐶, 𝑤} = {𝐶, 𝐴})
4241eqeq2d 2620 . . . . . . 7 (𝑤 = 𝐴 → ((𝐺 NeighbVtx 𝐴) = {𝐶, 𝑤} ↔ (𝐺 NeighbVtx 𝐴) = {𝐶, 𝐴}))
43 preq2 4213 . . . . . . . 8 (𝑤 = 𝐵 → {𝐶, 𝑤} = {𝐶, 𝐵})
4443eqeq2d 2620 . . . . . . 7 (𝑤 = 𝐵 → ((𝐺 NeighbVtx 𝐴) = {𝐶, 𝑤} ↔ (𝐺 NeighbVtx 𝐴) = {𝐶, 𝐵}))
4542, 44rexprg 4182 . . . . . 6 ((𝐴𝑋𝐵𝑌) → (∃𝑤 ∈ {𝐴, 𝐵} (𝐺 NeighbVtx 𝐴) = {𝐶, 𝑤} ↔ ((𝐺 NeighbVtx 𝐴) = {𝐶, 𝐴} ∨ (𝐺 NeighbVtx 𝐴) = {𝐶, 𝐵})))
46453adant3 1074 . . . . 5 ((𝐴𝑋𝐵𝑌𝐶𝑍) → (∃𝑤 ∈ {𝐴, 𝐵} (𝐺 NeighbVtx 𝐴) = {𝐶, 𝑤} ↔ ((𝐺 NeighbVtx 𝐴) = {𝐶, 𝐴} ∨ (𝐺 NeighbVtx 𝐴) = {𝐶, 𝐵})))
4733, 40, 463orbi123d 1390 . . . 4 ((𝐴𝑋𝐵𝑌𝐶𝑍) → ((∃𝑤 ∈ {𝐵, 𝐶} (𝐺 NeighbVtx 𝐴) = {𝐴, 𝑤} ∨ ∃𝑤 ∈ {𝐶, 𝐴} (𝐺 NeighbVtx 𝐴) = {𝐵, 𝑤} ∨ ∃𝑤 ∈ {𝐴, 𝐵} (𝐺 NeighbVtx 𝐴) = {𝐶, 𝑤}) ↔ (((𝐺 NeighbVtx 𝐴) = {𝐴, 𝐵} ∨ (𝐺 NeighbVtx 𝐴) = {𝐴, 𝐶}) ∨ ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ∨ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐴}) ∨ ((𝐺 NeighbVtx 𝐴) = {𝐶, 𝐴} ∨ (𝐺 NeighbVtx 𝐴) = {𝐶, 𝐵}))))
481, 47syl 17 . . 3 (𝜑 → ((∃𝑤 ∈ {𝐵, 𝐶} (𝐺 NeighbVtx 𝐴) = {𝐴, 𝑤} ∨ ∃𝑤 ∈ {𝐶, 𝐴} (𝐺 NeighbVtx 𝐴) = {𝐵, 𝑤} ∨ ∃𝑤 ∈ {𝐴, 𝐵} (𝐺 NeighbVtx 𝐴) = {𝐶, 𝑤}) ↔ (((𝐺 NeighbVtx 𝐴) = {𝐴, 𝐵} ∨ (𝐺 NeighbVtx 𝐴) = {𝐴, 𝐶}) ∨ ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ∨ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐴}) ∨ ((𝐺 NeighbVtx 𝐴) = {𝐶, 𝐴} ∨ (𝐺 NeighbVtx 𝐴) = {𝐶, 𝐵}))))
49 nb3grpr.n . . . 4 (𝜑 → (𝐴𝐵𝐴𝐶𝐵𝐶))
50 tprot 4228 . . . . . . . . 9 {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴}
5150a1i 11 . . . . . . . 8 ((𝐴𝐵𝐴𝐶𝐵𝐶) → {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴})
5251difeq1d 3689 . . . . . . 7 ((𝐴𝐵𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐴}) = ({𝐵, 𝐶, 𝐴} ∖ {𝐴}))
53 necom 2835 . . . . . . . . 9 (𝐴𝐵𝐵𝐴)
54 necom 2835 . . . . . . . . 9 (𝐴𝐶𝐶𝐴)
55 diftpsn3 4273 . . . . . . . . 9 ((𝐵𝐴𝐶𝐴) → ({𝐵, 𝐶, 𝐴} ∖ {𝐴}) = {𝐵, 𝐶})
5653, 54, 55syl2anb 495 . . . . . . . 8 ((𝐴𝐵𝐴𝐶) → ({𝐵, 𝐶, 𝐴} ∖ {𝐴}) = {𝐵, 𝐶})
57563adant3 1074 . . . . . . 7 ((𝐴𝐵𝐴𝐶𝐵𝐶) → ({𝐵, 𝐶, 𝐴} ∖ {𝐴}) = {𝐵, 𝐶})
5852, 57eqtrd 2644 . . . . . 6 ((𝐴𝐵𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐴}) = {𝐵, 𝐶})
5958rexeqdv 3122 . . . . 5 ((𝐴𝐵𝐴𝐶𝐵𝐶) → (∃𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴})(𝐺 NeighbVtx 𝐴) = {𝐴, 𝑤} ↔ ∃𝑤 ∈ {𝐵, 𝐶} (𝐺 NeighbVtx 𝐴) = {𝐴, 𝑤}))
60 tprot 4228 . . . . . . . . . 10 {𝐶, 𝐴, 𝐵} = {𝐴, 𝐵, 𝐶}
6160eqcomi 2619 . . . . . . . . 9 {𝐴, 𝐵, 𝐶} = {𝐶, 𝐴, 𝐵}
6261a1i 11 . . . . . . . 8 ((𝐴𝐵𝐴𝐶𝐵𝐶) → {𝐴, 𝐵, 𝐶} = {𝐶, 𝐴, 𝐵})
6362difeq1d 3689 . . . . . . 7 ((𝐴𝐵𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐵}) = ({𝐶, 𝐴, 𝐵} ∖ {𝐵}))
64 necom 2835 . . . . . . . . . . . 12 (𝐵𝐶𝐶𝐵)
6564anbi1i 727 . . . . . . . . . . 11 ((𝐵𝐶𝐴𝐵) ↔ (𝐶𝐵𝐴𝐵))
6665biimpi 205 . . . . . . . . . 10 ((𝐵𝐶𝐴𝐵) → (𝐶𝐵𝐴𝐵))
6766ancoms 468 . . . . . . . . 9 ((𝐴𝐵𝐵𝐶) → (𝐶𝐵𝐴𝐵))
68 diftpsn3 4273 . . . . . . . . 9 ((𝐶𝐵𝐴𝐵) → ({𝐶, 𝐴, 𝐵} ∖ {𝐵}) = {𝐶, 𝐴})
6967, 68syl 17 . . . . . . . 8 ((𝐴𝐵𝐵𝐶) → ({𝐶, 𝐴, 𝐵} ∖ {𝐵}) = {𝐶, 𝐴})
70693adant2 1073 . . . . . . 7 ((𝐴𝐵𝐴𝐶𝐵𝐶) → ({𝐶, 𝐴, 𝐵} ∖ {𝐵}) = {𝐶, 𝐴})
7163, 70eqtrd 2644 . . . . . 6 ((𝐴𝐵𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐵}) = {𝐶, 𝐴})
7271rexeqdv 3122 . . . . 5 ((𝐴𝐵𝐴𝐶𝐵𝐶) → (∃𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵})(𝐺 NeighbVtx 𝐴) = {𝐵, 𝑤} ↔ ∃𝑤 ∈ {𝐶, 𝐴} (𝐺 NeighbVtx 𝐴) = {𝐵, 𝑤}))
73 diftpsn3 4273 . . . . . . 7 ((𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐶}) = {𝐴, 𝐵})
74733adant1 1072 . . . . . 6 ((𝐴𝐵𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐶}) = {𝐴, 𝐵})
7574rexeqdv 3122 . . . . 5 ((𝐴𝐵𝐴𝐶𝐵𝐶) → (∃𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶})(𝐺 NeighbVtx 𝐴) = {𝐶, 𝑤} ↔ ∃𝑤 ∈ {𝐴, 𝐵} (𝐺 NeighbVtx 𝐴) = {𝐶, 𝑤}))
7659, 72, 753orbi123d 1390 . . . 4 ((𝐴𝐵𝐴𝐶𝐵𝐶) → ((∃𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴})(𝐺 NeighbVtx 𝐴) = {𝐴, 𝑤} ∨ ∃𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵})(𝐺 NeighbVtx 𝐴) = {𝐵, 𝑤} ∨ ∃𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶})(𝐺 NeighbVtx 𝐴) = {𝐶, 𝑤}) ↔ (∃𝑤 ∈ {𝐵, 𝐶} (𝐺 NeighbVtx 𝐴) = {𝐴, 𝑤} ∨ ∃𝑤 ∈ {𝐶, 𝐴} (𝐺 NeighbVtx 𝐴) = {𝐵, 𝑤} ∨ ∃𝑤 ∈ {𝐴, 𝐵} (𝐺 NeighbVtx 𝐴) = {𝐶, 𝑤})))
7749, 76syl 17 . . 3 (𝜑 → ((∃𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴})(𝐺 NeighbVtx 𝐴) = {𝐴, 𝑤} ∨ ∃𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵})(𝐺 NeighbVtx 𝐴) = {𝐵, 𝑤} ∨ ∃𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶})(𝐺 NeighbVtx 𝐴) = {𝐶, 𝑤}) ↔ (∃𝑤 ∈ {𝐵, 𝐶} (𝐺 NeighbVtx 𝐴) = {𝐴, 𝑤} ∨ ∃𝑤 ∈ {𝐶, 𝐴} (𝐺 NeighbVtx 𝐴) = {𝐵, 𝑤} ∨ ∃𝑤 ∈ {𝐴, 𝐵} (𝐺 NeighbVtx 𝐴) = {𝐶, 𝑤})))
78 prcom 4211 . . . . . . . 8 {𝐶, 𝐵} = {𝐵, 𝐶}
7978eqeq2i 2622 . . . . . . 7 ((𝐺 NeighbVtx 𝐴) = {𝐶, 𝐵} ↔ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶})
8079orbi2i 540 . . . . . 6 (((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ∨ (𝐺 NeighbVtx 𝐴) = {𝐶, 𝐵}) ↔ ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ∨ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}))
81 oridm 535 . . . . . 6 (((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ∨ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) ↔ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶})
8280, 81bitr2i 264 . . . . 5 ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ↔ ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ∨ (𝐺 NeighbVtx 𝐴) = {𝐶, 𝐵}))
8382a1i 11 . . . 4 (𝜑 → ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ↔ ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ∨ (𝐺 NeighbVtx 𝐴) = {𝐶, 𝐵})))
84 usgrnbnself2 40588 . . . . . . . 8 (𝐺 ∈ USGraph → 𝐴 ∉ (𝐺 NeighbVtx 𝐴))
85 df-nel 2783 . . . . . . . . 9 (𝐴 ∉ (𝐺 NeighbVtx 𝐴) ↔ ¬ 𝐴 ∈ (𝐺 NeighbVtx 𝐴))
86 prid2g 4240 . . . . . . . . . . . 12 (𝐴𝑋𝐴 ∈ {𝐵, 𝐴})
87863ad2ant1 1075 . . . . . . . . . . 11 ((𝐴𝑋𝐵𝑌𝐶𝑍) → 𝐴 ∈ {𝐵, 𝐴})
88 eleq2 2677 . . . . . . . . . . 11 ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐴} → (𝐴 ∈ (𝐺 NeighbVtx 𝐴) ↔ 𝐴 ∈ {𝐵, 𝐴}))
8987, 88syl5ibrcom 236 . . . . . . . . . 10 ((𝐴𝑋𝐵𝑌𝐶𝑍) → ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐴} → 𝐴 ∈ (𝐺 NeighbVtx 𝐴)))
9089con3rr3 150 . . . . . . . . 9 𝐴 ∈ (𝐺 NeighbVtx 𝐴) → ((𝐴𝑋𝐵𝑌𝐶𝑍) → ¬ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐴}))
9185, 90sylbi 206 . . . . . . . 8 (𝐴 ∉ (𝐺 NeighbVtx 𝐴) → ((𝐴𝑋𝐵𝑌𝐶𝑍) → ¬ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐴}))
9284, 91syl 17 . . . . . . 7 (𝐺 ∈ USGraph → ((𝐴𝑋𝐵𝑌𝐶𝑍) → ¬ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐴}))
9320, 1, 92sylc 63 . . . . . 6 (𝜑 → ¬ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐴})
94 biorf 419 . . . . . . 7 (¬ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐴} → ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ↔ ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐴} ∨ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶})))
95 orcom 401 . . . . . . 7 (((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐴} ∨ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) ↔ ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ∨ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐴}))
9694, 95syl6bb 275 . . . . . 6 (¬ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐴} → ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ↔ ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ∨ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐴})))
9793, 96syl 17 . . . . 5 (𝜑 → ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ↔ ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ∨ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐴})))
98 prid2g 4240 . . . . . . . . . . . 12 (𝐴𝑋𝐴 ∈ {𝐶, 𝐴})
99983ad2ant1 1075 . . . . . . . . . . 11 ((𝐴𝑋𝐵𝑌𝐶𝑍) → 𝐴 ∈ {𝐶, 𝐴})
100 eleq2 2677 . . . . . . . . . . 11 ((𝐺 NeighbVtx 𝐴) = {𝐶, 𝐴} → (𝐴 ∈ (𝐺 NeighbVtx 𝐴) ↔ 𝐴 ∈ {𝐶, 𝐴}))
10199, 100syl5ibrcom 236 . . . . . . . . . 10 ((𝐴𝑋𝐵𝑌𝐶𝑍) → ((𝐺 NeighbVtx 𝐴) = {𝐶, 𝐴} → 𝐴 ∈ (𝐺 NeighbVtx 𝐴)))
102101con3rr3 150 . . . . . . . . 9 𝐴 ∈ (𝐺 NeighbVtx 𝐴) → ((𝐴𝑋𝐵𝑌𝐶𝑍) → ¬ (𝐺 NeighbVtx 𝐴) = {𝐶, 𝐴}))
10385, 102sylbi 206 . . . . . . . 8 (𝐴 ∉ (𝐺 NeighbVtx 𝐴) → ((𝐴𝑋𝐵𝑌𝐶𝑍) → ¬ (𝐺 NeighbVtx 𝐴) = {𝐶, 𝐴}))
10484, 103syl 17 . . . . . . 7 (𝐺 ∈ USGraph → ((𝐴𝑋𝐵𝑌𝐶𝑍) → ¬ (𝐺 NeighbVtx 𝐴) = {𝐶, 𝐴}))
10520, 1, 104sylc 63 . . . . . 6 (𝜑 → ¬ (𝐺 NeighbVtx 𝐴) = {𝐶, 𝐴})
106 biorf 419 . . . . . 6 (¬ (𝐺 NeighbVtx 𝐴) = {𝐶, 𝐴} → ((𝐺 NeighbVtx 𝐴) = {𝐶, 𝐵} ↔ ((𝐺 NeighbVtx 𝐴) = {𝐶, 𝐴} ∨ (𝐺 NeighbVtx 𝐴) = {𝐶, 𝐵})))
107105, 106syl 17 . . . . 5 (𝜑 → ((𝐺 NeighbVtx 𝐴) = {𝐶, 𝐵} ↔ ((𝐺 NeighbVtx 𝐴) = {𝐶, 𝐴} ∨ (𝐺 NeighbVtx 𝐴) = {𝐶, 𝐵})))
10897, 107orbi12d 742 . . . 4 (𝜑 → (((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ∨ (𝐺 NeighbVtx 𝐴) = {𝐶, 𝐵}) ↔ (((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ∨ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐴}) ∨ ((𝐺 NeighbVtx 𝐴) = {𝐶, 𝐴} ∨ (𝐺 NeighbVtx 𝐴) = {𝐶, 𝐵}))))
109 prid1g 4239 . . . . . . . . . . . . . 14 (𝐴𝑋𝐴 ∈ {𝐴, 𝐵})
1101093ad2ant1 1075 . . . . . . . . . . . . 13 ((𝐴𝑋𝐵𝑌𝐶𝑍) → 𝐴 ∈ {𝐴, 𝐵})
111 eleq2 2677 . . . . . . . . . . . . 13 ((𝐺 NeighbVtx 𝐴) = {𝐴, 𝐵} → (𝐴 ∈ (𝐺 NeighbVtx 𝐴) ↔ 𝐴 ∈ {𝐴, 𝐵}))
112110, 111syl5ibrcom 236 . . . . . . . . . . . 12 ((𝐴𝑋𝐵𝑌𝐶𝑍) → ((𝐺 NeighbVtx 𝐴) = {𝐴, 𝐵} → 𝐴 ∈ (𝐺 NeighbVtx 𝐴)))
113112con3dimp 456 . . . . . . . . . . 11 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ ¬ 𝐴 ∈ (𝐺 NeighbVtx 𝐴)) → ¬ (𝐺 NeighbVtx 𝐴) = {𝐴, 𝐵})
114 prid1g 4239 . . . . . . . . . . . . . 14 (𝐴𝑋𝐴 ∈ {𝐴, 𝐶})
1151143ad2ant1 1075 . . . . . . . . . . . . 13 ((𝐴𝑋𝐵𝑌𝐶𝑍) → 𝐴 ∈ {𝐴, 𝐶})
116 eleq2 2677 . . . . . . . . . . . . 13 ((𝐺 NeighbVtx 𝐴) = {𝐴, 𝐶} → (𝐴 ∈ (𝐺 NeighbVtx 𝐴) ↔ 𝐴 ∈ {𝐴, 𝐶}))
117115, 116syl5ibrcom 236 . . . . . . . . . . . 12 ((𝐴𝑋𝐵𝑌𝐶𝑍) → ((𝐺 NeighbVtx 𝐴) = {𝐴, 𝐶} → 𝐴 ∈ (𝐺 NeighbVtx 𝐴)))
118117con3dimp 456 . . . . . . . . . . 11 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ ¬ 𝐴 ∈ (𝐺 NeighbVtx 𝐴)) → ¬ (𝐺 NeighbVtx 𝐴) = {𝐴, 𝐶})
119113, 118jca 553 . . . . . . . . . 10 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ ¬ 𝐴 ∈ (𝐺 NeighbVtx 𝐴)) → (¬ (𝐺 NeighbVtx 𝐴) = {𝐴, 𝐵} ∧ ¬ (𝐺 NeighbVtx 𝐴) = {𝐴, 𝐶}))
120119expcom 450 . . . . . . . . 9 𝐴 ∈ (𝐺 NeighbVtx 𝐴) → ((𝐴𝑋𝐵𝑌𝐶𝑍) → (¬ (𝐺 NeighbVtx 𝐴) = {𝐴, 𝐵} ∧ ¬ (𝐺 NeighbVtx 𝐴) = {𝐴, 𝐶})))
12185, 120sylbi 206 . . . . . . . 8 (𝐴 ∉ (𝐺 NeighbVtx 𝐴) → ((𝐴𝑋𝐵𝑌𝐶𝑍) → (¬ (𝐺 NeighbVtx 𝐴) = {𝐴, 𝐵} ∧ ¬ (𝐺 NeighbVtx 𝐴) = {𝐴, 𝐶})))
12284, 121syl 17 . . . . . . 7 (𝐺 ∈ USGraph → ((𝐴𝑋𝐵𝑌𝐶𝑍) → (¬ (𝐺 NeighbVtx 𝐴) = {𝐴, 𝐵} ∧ ¬ (𝐺 NeighbVtx 𝐴) = {𝐴, 𝐶})))
12320, 1, 122sylc 63 . . . . . 6 (𝜑 → (¬ (𝐺 NeighbVtx 𝐴) = {𝐴, 𝐵} ∧ ¬ (𝐺 NeighbVtx 𝐴) = {𝐴, 𝐶}))
124 ioran 510 . . . . . 6 (¬ ((𝐺 NeighbVtx 𝐴) = {𝐴, 𝐵} ∨ (𝐺 NeighbVtx 𝐴) = {𝐴, 𝐶}) ↔ (¬ (𝐺 NeighbVtx 𝐴) = {𝐴, 𝐵} ∧ ¬ (𝐺 NeighbVtx 𝐴) = {𝐴, 𝐶}))
125123, 124sylibr 223 . . . . 5 (𝜑 → ¬ ((𝐺 NeighbVtx 𝐴) = {𝐴, 𝐵} ∨ (𝐺 NeighbVtx 𝐴) = {𝐴, 𝐶}))
1261253bior1fd 1430 . . . 4 (𝜑 → ((((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ∨ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐴}) ∨ ((𝐺 NeighbVtx 𝐴) = {𝐶, 𝐴} ∨ (𝐺 NeighbVtx 𝐴) = {𝐶, 𝐵})) ↔ (((𝐺 NeighbVtx 𝐴) = {𝐴, 𝐵} ∨ (𝐺 NeighbVtx 𝐴) = {𝐴, 𝐶}) ∨ ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ∨ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐴}) ∨ ((𝐺 NeighbVtx 𝐴) = {𝐶, 𝐴} ∨ (𝐺 NeighbVtx 𝐴) = {𝐶, 𝐵}))))
12783, 108, 1263bitrd 293 . . 3 (𝜑 → ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ↔ (((𝐺 NeighbVtx 𝐴) = {𝐴, 𝐵} ∨ (𝐺 NeighbVtx 𝐴) = {𝐴, 𝐶}) ∨ ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ∨ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐴}) ∨ ((𝐺 NeighbVtx 𝐴) = {𝐶, 𝐴} ∨ (𝐺 NeighbVtx 𝐴) = {𝐶, 𝐵}))))
12848, 77, 1273bitr4rd 300 . 2 (𝜑 → ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ↔ (∃𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴})(𝐺 NeighbVtx 𝐴) = {𝐴, 𝑤} ∨ ∃𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵})(𝐺 NeighbVtx 𝐴) = {𝐵, 𝑤} ∨ ∃𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶})(𝐺 NeighbVtx 𝐴) = {𝐶, 𝑤})))
12918, 27, 1283bitr4rd 300 1 (𝜑 → ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ↔ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣})(𝐺 NeighbVtx 𝐴) = {𝑣, 𝑤}))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∨ w3o 1030   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   ∉ wnel 2781  ∃wrex 2897   ∖ cdif 3537  {csn 4125  {cpr 4127  {ctp 4129  ‘cfv 5804  (class class class)co 6549  Vtxcvtx 25673  Edgcedga 25792   USGraph cusgr 40379   NeighbVtx cnbgr 40550 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-8 1979  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-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  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-nel 2783  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  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-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-nbgr 40554 This theorem is referenced by:  nb3grpr  40610
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