Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  nb3grprlem1 Structured version   Visualization version   GIF version

Theorem nb3grprlem1 40608
 Description: Lemma 1 for nb3grapr 25982. (Contributed by Alexander van der Vekens, 15-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))
Assertion
Ref Expression
nb3grprlem1 (𝜑 → ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)))

Proof of Theorem nb3grprlem1
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 nb3grpr.s . . . . . . 7 (𝜑 → (𝐴𝑋𝐵𝑌𝐶𝑍))
2 prid1g 4239 . . . . . . . 8 (𝐵𝑌𝐵 ∈ {𝐵, 𝐶})
323ad2ant2 1076 . . . . . . 7 ((𝐴𝑋𝐵𝑌𝐶𝑍) → 𝐵 ∈ {𝐵, 𝐶})
41, 3syl 17 . . . . . 6 (𝜑𝐵 ∈ {𝐵, 𝐶})
54adantr 480 . . . . 5 ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → 𝐵 ∈ {𝐵, 𝐶})
6 eleq2 2677 . . . . . . 7 ({𝐵, 𝐶} = (𝐺 NeighbVtx 𝐴) → (𝐵 ∈ {𝐵, 𝐶} ↔ 𝐵 ∈ (𝐺 NeighbVtx 𝐴)))
76eqcoms 2618 . . . . . 6 ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} → (𝐵 ∈ {𝐵, 𝐶} ↔ 𝐵 ∈ (𝐺 NeighbVtx 𝐴)))
87adantl 481 . . . . 5 ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → (𝐵 ∈ {𝐵, 𝐶} ↔ 𝐵 ∈ (𝐺 NeighbVtx 𝐴)))
95, 8mpbid 221 . . . 4 ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → 𝐵 ∈ (𝐺 NeighbVtx 𝐴))
10 nb3grpr.g . . . . . 6 (𝜑𝐺 ∈ USGraph )
11 nb3grpr.e . . . . . . . 8 𝐸 = (Edg‘𝐺)
1211nbusgreledg 40575 . . . . . . 7 (𝐺 ∈ USGraph → (𝐵 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐵, 𝐴} ∈ 𝐸))
13 prcom 4211 . . . . . . . . 9 {𝐵, 𝐴} = {𝐴, 𝐵}
1413a1i 11 . . . . . . . 8 (𝐺 ∈ USGraph → {𝐵, 𝐴} = {𝐴, 𝐵})
1514eleq1d 2672 . . . . . . 7 (𝐺 ∈ USGraph → ({𝐵, 𝐴} ∈ 𝐸 ↔ {𝐴, 𝐵} ∈ 𝐸))
1612, 15bitrd 267 . . . . . 6 (𝐺 ∈ USGraph → (𝐵 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐴, 𝐵} ∈ 𝐸))
1710, 16syl 17 . . . . 5 (𝜑 → (𝐵 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐴, 𝐵} ∈ 𝐸))
1817adantr 480 . . . 4 ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → (𝐵 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐴, 𝐵} ∈ 𝐸))
199, 18mpbid 221 . . 3 ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → {𝐴, 𝐵} ∈ 𝐸)
20 prid2g 4240 . . . . . . . 8 (𝐶𝑍𝐶 ∈ {𝐵, 𝐶})
21203ad2ant3 1077 . . . . . . 7 ((𝐴𝑋𝐵𝑌𝐶𝑍) → 𝐶 ∈ {𝐵, 𝐶})
221, 21syl 17 . . . . . 6 (𝜑𝐶 ∈ {𝐵, 𝐶})
2322adantr 480 . . . . 5 ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → 𝐶 ∈ {𝐵, 𝐶})
24 eleq2 2677 . . . . . . 7 ({𝐵, 𝐶} = (𝐺 NeighbVtx 𝐴) → (𝐶 ∈ {𝐵, 𝐶} ↔ 𝐶 ∈ (𝐺 NeighbVtx 𝐴)))
2524eqcoms 2618 . . . . . 6 ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} → (𝐶 ∈ {𝐵, 𝐶} ↔ 𝐶 ∈ (𝐺 NeighbVtx 𝐴)))
2625adantl 481 . . . . 5 ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → (𝐶 ∈ {𝐵, 𝐶} ↔ 𝐶 ∈ (𝐺 NeighbVtx 𝐴)))
2723, 26mpbid 221 . . . 4 ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → 𝐶 ∈ (𝐺 NeighbVtx 𝐴))
2811nbusgreledg 40575 . . . . . . 7 (𝐺 ∈ USGraph → (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐶, 𝐴} ∈ 𝐸))
29 prcom 4211 . . . . . . . . 9 {𝐶, 𝐴} = {𝐴, 𝐶}
3029a1i 11 . . . . . . . 8 (𝐺 ∈ USGraph → {𝐶, 𝐴} = {𝐴, 𝐶})
3130eleq1d 2672 . . . . . . 7 (𝐺 ∈ USGraph → ({𝐶, 𝐴} ∈ 𝐸 ↔ {𝐴, 𝐶} ∈ 𝐸))
3228, 31bitrd 267 . . . . . 6 (𝐺 ∈ USGraph → (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐴, 𝐶} ∈ 𝐸))
3310, 32syl 17 . . . . 5 (𝜑 → (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐴, 𝐶} ∈ 𝐸))
3433adantr 480 . . . 4 ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐴, 𝐶} ∈ 𝐸))
3527, 34mpbid 221 . . 3 ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → {𝐴, 𝐶} ∈ 𝐸)
3619, 35jca 553 . 2 ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸))
37 nb3grpr.v . . . . . 6 𝑉 = (Vtx‘𝐺)
3837, 11nbusgr 40571 . . . . 5 (𝐺 ∈ USGraph → (𝐺 NeighbVtx 𝐴) = {𝑣𝑉 ∣ {𝐴, 𝑣} ∈ 𝐸})
3910, 38syl 17 . . . 4 (𝜑 → (𝐺 NeighbVtx 𝐴) = {𝑣𝑉 ∣ {𝐴, 𝑣} ∈ 𝐸})
4039adantr 480 . . 3 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝐺 NeighbVtx 𝐴) = {𝑣𝑉 ∣ {𝐴, 𝑣} ∈ 𝐸})
41 nb3grpr.t . . . . . . . . . 10 (𝜑𝑉 = {𝐴, 𝐵, 𝐶})
42 eleq2 2677 . . . . . . . . . 10 (𝑉 = {𝐴, 𝐵, 𝐶} → (𝑣𝑉𝑣 ∈ {𝐴, 𝐵, 𝐶}))
4341, 42syl 17 . . . . . . . . 9 (𝜑 → (𝑣𝑉𝑣 ∈ {𝐴, 𝐵, 𝐶}))
4443adantr 480 . . . . . . . 8 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣𝑉𝑣 ∈ {𝐴, 𝐵, 𝐶}))
45 vex 3176 . . . . . . . . . . 11 𝑣 ∈ V
4645eltp 4177 . . . . . . . . . 10 (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝑣 = 𝐴𝑣 = 𝐵𝑣 = 𝐶))
4711usgredgne 40433 . . . . . . . . . . . . . . . 16 ((𝐺 ∈ USGraph ∧ {𝐴, 𝑣} ∈ 𝐸) → 𝐴𝑣)
48 df-ne 2782 . . . . . . . . . . . . . . . . 17 (𝐴𝑣 ↔ ¬ 𝐴 = 𝑣)
49 pm2.24 120 . . . . . . . . . . . . . . . . . . 19 (𝐴 = 𝑣 → (¬ 𝐴 = 𝑣 → (𝑣 = 𝐵𝑣 = 𝐶)))
5049eqcoms 2618 . . . . . . . . . . . . . . . . . 18 (𝑣 = 𝐴 → (¬ 𝐴 = 𝑣 → (𝑣 = 𝐵𝑣 = 𝐶)))
5150com12 32 . . . . . . . . . . . . . . . . 17 𝐴 = 𝑣 → (𝑣 = 𝐴 → (𝑣 = 𝐵𝑣 = 𝐶)))
5248, 51sylbi 206 . . . . . . . . . . . . . . . 16 (𝐴𝑣 → (𝑣 = 𝐴 → (𝑣 = 𝐵𝑣 = 𝐶)))
5347, 52syl 17 . . . . . . . . . . . . . . 15 ((𝐺 ∈ USGraph ∧ {𝐴, 𝑣} ∈ 𝐸) → (𝑣 = 𝐴 → (𝑣 = 𝐵𝑣 = 𝐶)))
5453ex 449 . . . . . . . . . . . . . 14 (𝐺 ∈ USGraph → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐴 → (𝑣 = 𝐵𝑣 = 𝐶))))
5510, 54syl 17 . . . . . . . . . . . . 13 (𝜑 → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐴 → (𝑣 = 𝐵𝑣 = 𝐶))))
5655adantr 480 . . . . . . . . . . . 12 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐴 → (𝑣 = 𝐵𝑣 = 𝐶))))
5756com3r 85 . . . . . . . . . . 11 (𝑣 = 𝐴 → ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐵𝑣 = 𝐶))))
58 orc 399 . . . . . . . . . . . 12 (𝑣 = 𝐵 → (𝑣 = 𝐵𝑣 = 𝐶))
59582a1d 26 . . . . . . . . . . 11 (𝑣 = 𝐵 → ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐵𝑣 = 𝐶))))
60 olc 398 . . . . . . . . . . . 12 (𝑣 = 𝐶 → (𝑣 = 𝐵𝑣 = 𝐶))
61602a1d 26 . . . . . . . . . . 11 (𝑣 = 𝐶 → ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐵𝑣 = 𝐶))))
6257, 59, 613jaoi 1383 . . . . . . . . . 10 ((𝑣 = 𝐴𝑣 = 𝐵𝑣 = 𝐶) → ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐵𝑣 = 𝐶))))
6346, 62sylbi 206 . . . . . . . . 9 (𝑣 ∈ {𝐴, 𝐵, 𝐶} → ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐵𝑣 = 𝐶))))
6463com12 32 . . . . . . . 8 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣 ∈ {𝐴, 𝐵, 𝐶} → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐵𝑣 = 𝐶))))
6544, 64sylbid 229 . . . . . . 7 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣𝑉 → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐵𝑣 = 𝐶))))
6665impd 446 . . . . . 6 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ((𝑣𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸) → (𝑣 = 𝐵𝑣 = 𝐶)))
67 eqid 2610 . . . . . . . . . . . . . . . . . 18 𝐵 = 𝐵
68673mix2i 1227 . . . . . . . . . . . . . . . . 17 (𝐵 = 𝐴𝐵 = 𝐵𝐵 = 𝐶)
691simp2d 1067 . . . . . . . . . . . . . . . . . 18 (𝜑𝐵𝑌)
70 eltpg 4174 . . . . . . . . . . . . . . . . . 18 (𝐵𝑌 → (𝐵 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝐵 = 𝐴𝐵 = 𝐵𝐵 = 𝐶)))
7169, 70syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐵 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝐵 = 𝐴𝐵 = 𝐵𝐵 = 𝐶)))
7268, 71mpbiri 247 . . . . . . . . . . . . . . . 16 (𝜑𝐵 ∈ {𝐴, 𝐵, 𝐶})
7372adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑣 = 𝐵) → 𝐵 ∈ {𝐴, 𝐵, 𝐶})
74 eleq1 2676 . . . . . . . . . . . . . . . . 17 (𝑣 = 𝐵 → (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝐵 ∈ {𝐴, 𝐵, 𝐶}))
7574bicomd 212 . . . . . . . . . . . . . . . 16 (𝑣 = 𝐵 → (𝐵 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣 ∈ {𝐴, 𝐵, 𝐶}))
7675adantl 481 . . . . . . . . . . . . . . 15 ((𝜑𝑣 = 𝐵) → (𝐵 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣 ∈ {𝐴, 𝐵, 𝐶}))
7773, 76mpbid 221 . . . . . . . . . . . . . 14 ((𝜑𝑣 = 𝐵) → 𝑣 ∈ {𝐴, 𝐵, 𝐶})
7842bicomd 212 . . . . . . . . . . . . . . . 16 (𝑉 = {𝐴, 𝐵, 𝐶} → (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣𝑉))
7941, 78syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣𝑉))
8079adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑣 = 𝐵) → (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣𝑉))
8177, 80mpbid 221 . . . . . . . . . . . . 13 ((𝜑𝑣 = 𝐵) → 𝑣𝑉)
8281ex 449 . . . . . . . . . . . 12 (𝜑 → (𝑣 = 𝐵𝑣𝑉))
8382adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣 = 𝐵𝑣𝑉))
8483impcom 445 . . . . . . . . . 10 ((𝑣 = 𝐵 ∧ (𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸))) → 𝑣𝑉)
85 preq2 4213 . . . . . . . . . . . . . . 15 (𝐵 = 𝑣 → {𝐴, 𝐵} = {𝐴, 𝑣})
8685eleq1d 2672 . . . . . . . . . . . . . 14 (𝐵 = 𝑣 → ({𝐴, 𝐵} ∈ 𝐸 ↔ {𝐴, 𝑣} ∈ 𝐸))
8786eqcoms 2618 . . . . . . . . . . . . 13 (𝑣 = 𝐵 → ({𝐴, 𝐵} ∈ 𝐸 ↔ {𝐴, 𝑣} ∈ 𝐸))
8887biimpcd 238 . . . . . . . . . . . 12 ({𝐴, 𝐵} ∈ 𝐸 → (𝑣 = 𝐵 → {𝐴, 𝑣} ∈ 𝐸))
8988ad2antrl 760 . . . . . . . . . . 11 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣 = 𝐵 → {𝐴, 𝑣} ∈ 𝐸))
9089impcom 445 . . . . . . . . . 10 ((𝑣 = 𝐵 ∧ (𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸))) → {𝐴, 𝑣} ∈ 𝐸)
9184, 90jca 553 . . . . . . . . 9 ((𝑣 = 𝐵 ∧ (𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸))) → (𝑣𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸))
9291ex 449 . . . . . . . 8 (𝑣 = 𝐵 → ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸)))
93 tpid3g 4248 . . . . . . . . . . . . . . . . . 18 (𝐶𝑍𝐶 ∈ {𝐴, 𝐵, 𝐶})
94933ad2ant3 1077 . . . . . . . . . . . . . . . . 17 ((𝐴𝑋𝐵𝑌𝐶𝑍) → 𝐶 ∈ {𝐴, 𝐵, 𝐶})
951, 94syl 17 . . . . . . . . . . . . . . . 16 (𝜑𝐶 ∈ {𝐴, 𝐵, 𝐶})
9695adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑣 = 𝐶) → 𝐶 ∈ {𝐴, 𝐵, 𝐶})
97 eleq1 2676 . . . . . . . . . . . . . . . . 17 (𝑣 = 𝐶 → (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝐶 ∈ {𝐴, 𝐵, 𝐶}))
9897bicomd 212 . . . . . . . . . . . . . . . 16 (𝑣 = 𝐶 → (𝐶 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣 ∈ {𝐴, 𝐵, 𝐶}))
9998adantl 481 . . . . . . . . . . . . . . 15 ((𝜑𝑣 = 𝐶) → (𝐶 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣 ∈ {𝐴, 𝐵, 𝐶}))
10096, 99mpbid 221 . . . . . . . . . . . . . 14 ((𝜑𝑣 = 𝐶) → 𝑣 ∈ {𝐴, 𝐵, 𝐶})
10179adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑣 = 𝐶) → (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣𝑉))
102100, 101mpbid 221 . . . . . . . . . . . . 13 ((𝜑𝑣 = 𝐶) → 𝑣𝑉)
103102ex 449 . . . . . . . . . . . 12 (𝜑 → (𝑣 = 𝐶𝑣𝑉))
104103adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣 = 𝐶𝑣𝑉))
105104impcom 445 . . . . . . . . . 10 ((𝑣 = 𝐶 ∧ (𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸))) → 𝑣𝑉)
106 preq2 4213 . . . . . . . . . . . . . . 15 (𝐶 = 𝑣 → {𝐴, 𝐶} = {𝐴, 𝑣})
107106eleq1d 2672 . . . . . . . . . . . . . 14 (𝐶 = 𝑣 → ({𝐴, 𝐶} ∈ 𝐸 ↔ {𝐴, 𝑣} ∈ 𝐸))
108107eqcoms 2618 . . . . . . . . . . . . 13 (𝑣 = 𝐶 → ({𝐴, 𝐶} ∈ 𝐸 ↔ {𝐴, 𝑣} ∈ 𝐸))
109108biimpcd 238 . . . . . . . . . . . 12 ({𝐴, 𝐶} ∈ 𝐸 → (𝑣 = 𝐶 → {𝐴, 𝑣} ∈ 𝐸))
110109ad2antll 761 . . . . . . . . . . 11 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣 = 𝐶 → {𝐴, 𝑣} ∈ 𝐸))
111110impcom 445 . . . . . . . . . 10 ((𝑣 = 𝐶 ∧ (𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸))) → {𝐴, 𝑣} ∈ 𝐸)
112105, 111jca 553 . . . . . . . . 9 ((𝑣 = 𝐶 ∧ (𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸))) → (𝑣𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸))
113112ex 449 . . . . . . . 8 (𝑣 = 𝐶 → ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸)))
11492, 113jaoi 393 . . . . . . 7 ((𝑣 = 𝐵𝑣 = 𝐶) → ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸)))
115114com12 32 . . . . . 6 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ((𝑣 = 𝐵𝑣 = 𝐶) → (𝑣𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸)))
11666, 115impbid 201 . . . . 5 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ((𝑣𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸) ↔ (𝑣 = 𝐵𝑣 = 𝐶)))
117116abbidv 2728 . . . 4 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → {𝑣 ∣ (𝑣𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸)} = {𝑣 ∣ (𝑣 = 𝐵𝑣 = 𝐶)})
118 df-rab 2905 . . . 4 {𝑣𝑉 ∣ {𝐴, 𝑣} ∈ 𝐸} = {𝑣 ∣ (𝑣𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸)}
119 dfpr2 4143 . . . 4 {𝐵, 𝐶} = {𝑣 ∣ (𝑣 = 𝐵𝑣 = 𝐶)}
120117, 118, 1193eqtr4g 2669 . . 3 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → {𝑣𝑉 ∣ {𝐴, 𝑣} ∈ 𝐸} = {𝐵, 𝐶})
12140, 120eqtrd 2644 . 2 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶})
12236, 121impbida 873 1 (𝜑 → ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)))
 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  {cab 2596   ≠ wne 2780  {crab 2900  {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-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 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-reu 2903  df-rmo 2904  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-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  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-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-hash 12980  df-upgr 25749  df-umgr 25750  df-edga 25793  df-usgr 40381  df-nbgr 40554 This theorem is referenced by:  nb3grpr  40610  nb3grpr2  40611  nb3gr2nb  40612
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