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Theorem nb3grprlem1 26326
Description: Lemma 1 for nb3grpr 26328. (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 4327 . . . . . . . 8 (𝐵𝑌𝐵 ∈ {𝐵, 𝐶})
323ad2ant2 1103 . . . . . . 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 2719 . . . . . . 7 ({𝐵, 𝐶} = (𝐺 NeighbVtx 𝐴) → (𝐵 ∈ {𝐵, 𝐶} ↔ 𝐵 ∈ (𝐺 NeighbVtx 𝐴)))
76eqcoms 2659 . . . . . 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 222 . . . 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 26294 . . . . . . 7 (𝐺 ∈ USGraph → (𝐵 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐵, 𝐴} ∈ 𝐸))
13 prcom 4299 . . . . . . . . 9 {𝐵, 𝐴} = {𝐴, 𝐵}
1413a1i 11 . . . . . . . 8 (𝐺 ∈ USGraph → {𝐵, 𝐴} = {𝐴, 𝐵})
1514eleq1d 2715 . . . . . . 7 (𝐺 ∈ USGraph → ({𝐵, 𝐴} ∈ 𝐸 ↔ {𝐴, 𝐵} ∈ 𝐸))
1612, 15bitrd 268 . . . . . 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 222 . . 3 ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → {𝐴, 𝐵} ∈ 𝐸)
20 prid2g 4328 . . . . . . . 8 (𝐶𝑍𝐶 ∈ {𝐵, 𝐶})
21203ad2ant3 1104 . . . . . . 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 2719 . . . . . . 7 ({𝐵, 𝐶} = (𝐺 NeighbVtx 𝐴) → (𝐶 ∈ {𝐵, 𝐶} ↔ 𝐶 ∈ (𝐺 NeighbVtx 𝐴)))
2524eqcoms 2659 . . . . . 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 222 . . . 4 ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → 𝐶 ∈ (𝐺 NeighbVtx 𝐴))
2811nbusgreledg 26294 . . . . . . 7 (𝐺 ∈ USGraph → (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐶, 𝐴} ∈ 𝐸))
29 prcom 4299 . . . . . . . . 9 {𝐶, 𝐴} = {𝐴, 𝐶}
3029a1i 11 . . . . . . . 8 (𝐺 ∈ USGraph → {𝐶, 𝐴} = {𝐴, 𝐶})
3130eleq1d 2715 . . . . . . 7 (𝐺 ∈ USGraph → ({𝐶, 𝐴} ∈ 𝐸 ↔ {𝐴, 𝐶} ∈ 𝐸))
3228, 31bitrd 268 . . . . . 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 222 . . 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 26290 . . . . 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 2719 . . . . . . . . . 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 3234 . . . . . . . . . . 11 𝑣 ∈ V
4645eltp 4262 . . . . . . . . . 10 (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝑣 = 𝐴𝑣 = 𝐵𝑣 = 𝐶))
4711usgredgne 26143 . . . . . . . . . . . . . . . 16 ((𝐺 ∈ USGraph ∧ {𝐴, 𝑣} ∈ 𝐸) → 𝐴𝑣)
48 df-ne 2824 . . . . . . . . . . . . . . . . 17 (𝐴𝑣 ↔ ¬ 𝐴 = 𝑣)
49 pm2.24 121 . . . . . . . . . . . . . . . . . . 19 (𝐴 = 𝑣 → (¬ 𝐴 = 𝑣 → (𝑣 = 𝐵𝑣 = 𝐶)))
5049eqcoms 2659 . . . . . . . . . . . . . . . . . 18 (𝑣 = 𝐴 → (¬ 𝐴 = 𝑣 → (𝑣 = 𝐵𝑣 = 𝐶)))
5150com12 32 . . . . . . . . . . . . . . . . 17 𝐴 = 𝑣 → (𝑣 = 𝐴 → (𝑣 = 𝐵𝑣 = 𝐶)))
5248, 51sylbi 207 . . . . . . . . . . . . . . . 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 87 . . . . . . . . . . 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 1431 . . . . . . . . . 10 ((𝑣 = 𝐴𝑣 = 𝐵𝑣 = 𝐶) → ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐵𝑣 = 𝐶))))
6346, 62sylbi 207 . . . . . . . . 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 230 . . . . . . 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 2651 . . . . . . . . . . . . . . . . . 18 𝐵 = 𝐵
68673mix2i 1254 . . . . . . . . . . . . . . . . 17 (𝐵 = 𝐴𝐵 = 𝐵𝐵 = 𝐶)
691simp2d 1094 . . . . . . . . . . . . . . . . . 18 (𝜑𝐵𝑌)
70 eltpg 4259 . . . . . . . . . . . . . . . . . 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 248 . . . . . . . . . . . . . . . 16 (𝜑𝐵 ∈ {𝐴, 𝐵, 𝐶})
7372adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑣 = 𝐵) → 𝐵 ∈ {𝐴, 𝐵, 𝐶})
74 eleq1 2718 . . . . . . . . . . . . . . . . 17 (𝑣 = 𝐵 → (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝐵 ∈ {𝐴, 𝐵, 𝐶}))
7574bicomd 213 . . . . . . . . . . . . . . . 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 222 . . . . . . . . . . . . . 14 ((𝜑𝑣 = 𝐵) → 𝑣 ∈ {𝐴, 𝐵, 𝐶})
7842bicomd 213 . . . . . . . . . . . . . . . 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 222 . . . . . . . . . . . . 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 4301 . . . . . . . . . . . . . . 15 (𝐵 = 𝑣 → {𝐴, 𝐵} = {𝐴, 𝑣})
8685eleq1d 2715 . . . . . . . . . . . . . 14 (𝐵 = 𝑣 → ({𝐴, 𝐵} ∈ 𝐸 ↔ {𝐴, 𝑣} ∈ 𝐸))
8786eqcoms 2659 . . . . . . . . . . . . 13 (𝑣 = 𝐵 → ({𝐴, 𝐵} ∈ 𝐸 ↔ {𝐴, 𝑣} ∈ 𝐸))
8887biimpcd 239 . . . . . . . . . . . 12 ({𝐴, 𝐵} ∈ 𝐸 → (𝑣 = 𝐵 → {𝐴, 𝑣} ∈ 𝐸))
8988ad2antrl 764 . . . . . . . . . . 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 4337 . . . . . . . . . . . . . . . . . 18 (𝐶𝑍𝐶 ∈ {𝐴, 𝐵, 𝐶})
94933ad2ant3 1104 . . . . . . . . . . . . . . . . 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 2718 . . . . . . . . . . . . . . . . 17 (𝑣 = 𝐶 → (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝐶 ∈ {𝐴, 𝐵, 𝐶}))
9897bicomd 213 . . . . . . . . . . . . . . . 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 222 . . . . . . . . . . . . . 14 ((𝜑𝑣 = 𝐶) → 𝑣 ∈ {𝐴, 𝐵, 𝐶})
10179adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑣 = 𝐶) → (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣𝑉))
102100, 101mpbid 222 . . . . . . . . . . . . 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 4301 . . . . . . . . . . . . . . 15 (𝐶 = 𝑣 → {𝐴, 𝐶} = {𝐴, 𝑣})
107106eleq1d 2715 . . . . . . . . . . . . . 14 (𝐶 = 𝑣 → ({𝐴, 𝐶} ∈ 𝐸 ↔ {𝐴, 𝑣} ∈ 𝐸))
108107eqcoms 2659 . . . . . . . . . . . . 13 (𝑣 = 𝐶 → ({𝐴, 𝐶} ∈ 𝐸 ↔ {𝐴, 𝑣} ∈ 𝐸))
109108biimpcd 239 . . . . . . . . . . . 12 ({𝐴, 𝐶} ∈ 𝐸 → (𝑣 = 𝐶 → {𝐴, 𝑣} ∈ 𝐸))
110109ad2antll 765 . . . . . . . . . . 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 202 . . . . 5 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ((𝑣𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸) ↔ (𝑣 = 𝐵𝑣 = 𝐶)))
117116abbidv 2770 . . . 4 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → {𝑣 ∣ (𝑣𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸)} = {𝑣 ∣ (𝑣 = 𝐵𝑣 = 𝐶)})
118 df-rab 2950 . . . 4 {𝑣𝑉 ∣ {𝐴, 𝑣} ∈ 𝐸} = {𝑣 ∣ (𝑣𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸)}
119 dfpr2 4228 . . . 4 {𝐵, 𝐶} = {𝑣 ∣ (𝑣 = 𝐵𝑣 = 𝐶)}
120117, 118, 1193eqtr4g 2710 . . 3 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → {𝑣𝑉 ∣ {𝐴, 𝑣} ∈ 𝐸} = {𝐵, 𝐶})
12140, 120eqtrd 2685 . 2 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶})
12236, 121impbida 895 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 196  wo 382  wa 383  w3o 1053  w3a 1054   = wceq 1523  wcel 2030  {cab 2637  wne 2823  {crab 2945  {cpr 4212  {ctp 4214  cfv 5926  (class class class)co 6690  Vtxcvtx 25919  Edgcedg 25984  USGraphcusgr 26089   NeighbVtx cnbgr 26269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-n0 11331  df-xnn0 11402  df-z 11416  df-uz 11726  df-fz 12365  df-hash 13158  df-edg 25985  df-upgr 26022  df-umgr 26023  df-usgr 26091  df-nbgr 26270
This theorem is referenced by:  nb3grpr  26328  nb3grpr2  26329  nb3gr2nb  26330
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