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Theorem nb3graprlem1 25980
 Description: Lemma 1 for nb3grapr 25982. (Contributed by Alexander van der Vekens, 15-Oct-2017.)
Assertion
Ref Expression
nb3graprlem1 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) → ((⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝐵, 𝐶} ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸)))

Proof of Theorem nb3graprlem1
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 prid1g 4239 . . . . . . . 8 (𝐵𝑌𝐵 ∈ {𝐵, 𝐶})
213ad2ant2 1076 . . . . . . 7 ((𝐴𝑋𝐵𝑌𝐶𝑍) → 𝐵 ∈ {𝐵, 𝐶})
32adantr 480 . . . . . 6 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) → 𝐵 ∈ {𝐵, 𝐶})
43adantr 480 . . . . 5 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ (⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝐵, 𝐶}) → 𝐵 ∈ {𝐵, 𝐶})
5 eleq2 2677 . . . . . . 7 ({𝐵, 𝐶} = (⟨𝑉, 𝐸⟩ Neighbors 𝐴) → (𝐵 ∈ {𝐵, 𝐶} ↔ 𝐵 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝐴)))
65eqcoms 2618 . . . . . 6 ((⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝐵, 𝐶} → (𝐵 ∈ {𝐵, 𝐶} ↔ 𝐵 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝐴)))
76adantl 481 . . . . 5 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ (⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝐵, 𝐶}) → (𝐵 ∈ {𝐵, 𝐶} ↔ 𝐵 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝐴)))
84, 7mpbid 221 . . . 4 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ (⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝐵, 𝐶}) → 𝐵 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝐴))
9 nbgraeledg 25959 . . . . . . 7 (𝑉 USGrph 𝐸 → (𝐵 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝐴) ↔ {𝐵, 𝐴} ∈ ran 𝐸))
10 prcom 4211 . . . . . . . . 9 {𝐵, 𝐴} = {𝐴, 𝐵}
1110a1i 11 . . . . . . . 8 (𝑉 USGrph 𝐸 → {𝐵, 𝐴} = {𝐴, 𝐵})
1211eleq1d 2672 . . . . . . 7 (𝑉 USGrph 𝐸 → ({𝐵, 𝐴} ∈ ran 𝐸 ↔ {𝐴, 𝐵} ∈ ran 𝐸))
139, 12bitrd 267 . . . . . 6 (𝑉 USGrph 𝐸 → (𝐵 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝐴) ↔ {𝐴, 𝐵} ∈ ran 𝐸))
1413adantl 481 . . . . 5 ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸) → (𝐵 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝐴) ↔ {𝐴, 𝐵} ∈ ran 𝐸))
1514ad2antlr 759 . . . 4 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ (⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝐵, 𝐶}) → (𝐵 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝐴) ↔ {𝐴, 𝐵} ∈ ran 𝐸))
168, 15mpbid 221 . . 3 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ (⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝐵, 𝐶}) → {𝐴, 𝐵} ∈ ran 𝐸)
17 prid2g 4240 . . . . . . . 8 (𝐶𝑍𝐶 ∈ {𝐵, 𝐶})
18173ad2ant3 1077 . . . . . . 7 ((𝐴𝑋𝐵𝑌𝐶𝑍) → 𝐶 ∈ {𝐵, 𝐶})
1918adantr 480 . . . . . 6 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) → 𝐶 ∈ {𝐵, 𝐶})
2019adantr 480 . . . . 5 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ (⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝐵, 𝐶}) → 𝐶 ∈ {𝐵, 𝐶})
21 eleq2 2677 . . . . . . 7 ({𝐵, 𝐶} = (⟨𝑉, 𝐸⟩ Neighbors 𝐴) → (𝐶 ∈ {𝐵, 𝐶} ↔ 𝐶 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝐴)))
2221eqcoms 2618 . . . . . 6 ((⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝐵, 𝐶} → (𝐶 ∈ {𝐵, 𝐶} ↔ 𝐶 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝐴)))
2322adantl 481 . . . . 5 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ (⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝐵, 𝐶}) → (𝐶 ∈ {𝐵, 𝐶} ↔ 𝐶 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝐴)))
2420, 23mpbid 221 . . . 4 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ (⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝐵, 𝐶}) → 𝐶 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝐴))
25 nbgraeledg 25959 . . . . . . 7 (𝑉 USGrph 𝐸 → (𝐶 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝐴) ↔ {𝐶, 𝐴} ∈ ran 𝐸))
26 prcom 4211 . . . . . . . . 9 {𝐶, 𝐴} = {𝐴, 𝐶}
2726a1i 11 . . . . . . . 8 (𝑉 USGrph 𝐸 → {𝐶, 𝐴} = {𝐴, 𝐶})
2827eleq1d 2672 . . . . . . 7 (𝑉 USGrph 𝐸 → ({𝐶, 𝐴} ∈ ran 𝐸 ↔ {𝐴, 𝐶} ∈ ran 𝐸))
2925, 28bitrd 267 . . . . . 6 (𝑉 USGrph 𝐸 → (𝐶 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝐴) ↔ {𝐴, 𝐶} ∈ ran 𝐸))
3029adantl 481 . . . . 5 ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸) → (𝐶 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝐴) ↔ {𝐴, 𝐶} ∈ ran 𝐸))
3130ad2antlr 759 . . . 4 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ (⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝐵, 𝐶}) → (𝐶 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝐴) ↔ {𝐴, 𝐶} ∈ ran 𝐸))
3224, 31mpbid 221 . . 3 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ (⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝐵, 𝐶}) → {𝐴, 𝐶} ∈ ran 𝐸)
3316, 32jca 553 . 2 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ (⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝐵, 𝐶}) → ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸))
34 nbusgra 25957 . . . . 5 (𝑉 USGrph 𝐸 → (⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝑣𝑉 ∣ {𝐴, 𝑣} ∈ ran 𝐸})
3534ad2antll 761 . . . 4 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) → (⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝑣𝑉 ∣ {𝐴, 𝑣} ∈ ran 𝐸})
3635adantr 480 . . 3 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸)) → (⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝑣𝑉 ∣ {𝐴, 𝑣} ∈ ran 𝐸})
37 eleq2 2677 . . . . . . . . . 10 (𝑉 = {𝐴, 𝐵, 𝐶} → (𝑣𝑉𝑣 ∈ {𝐴, 𝐵, 𝐶}))
3837ad2antrl 760 . . . . . . . . 9 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) → (𝑣𝑉𝑣 ∈ {𝐴, 𝐵, 𝐶}))
3938adantr 480 . . . . . . . 8 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸)) → (𝑣𝑉𝑣 ∈ {𝐴, 𝐵, 𝐶}))
40 vex 3176 . . . . . . . . . . 11 𝑣 ∈ V
4140eltp 4177 . . . . . . . . . 10 (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝑣 = 𝐴𝑣 = 𝐵𝑣 = 𝐶))
42 usgraedgrn 25910 . . . . . . . . . . . . . . . 16 ((𝑉 USGrph 𝐸 ∧ {𝐴, 𝑣} ∈ ran 𝐸) → 𝐴𝑣)
43 df-ne 2782 . . . . . . . . . . . . . . . . 17 (𝐴𝑣 ↔ ¬ 𝐴 = 𝑣)
44 pm2.24 120 . . . . . . . . . . . . . . . . . . 19 (𝐴 = 𝑣 → (¬ 𝐴 = 𝑣 → (𝑣 = 𝐵𝑣 = 𝐶)))
4544eqcoms 2618 . . . . . . . . . . . . . . . . . 18 (𝑣 = 𝐴 → (¬ 𝐴 = 𝑣 → (𝑣 = 𝐵𝑣 = 𝐶)))
4645com12 32 . . . . . . . . . . . . . . . . 17 𝐴 = 𝑣 → (𝑣 = 𝐴 → (𝑣 = 𝐵𝑣 = 𝐶)))
4743, 46sylbi 206 . . . . . . . . . . . . . . . 16 (𝐴𝑣 → (𝑣 = 𝐴 → (𝑣 = 𝐵𝑣 = 𝐶)))
4842, 47syl 17 . . . . . . . . . . . . . . 15 ((𝑉 USGrph 𝐸 ∧ {𝐴, 𝑣} ∈ ran 𝐸) → (𝑣 = 𝐴 → (𝑣 = 𝐵𝑣 = 𝐶)))
4948ex 449 . . . . . . . . . . . . . 14 (𝑉 USGrph 𝐸 → ({𝐴, 𝑣} ∈ ran 𝐸 → (𝑣 = 𝐴 → (𝑣 = 𝐵𝑣 = 𝐶))))
5049ad2antll 761 . . . . . . . . . . . . 13 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) → ({𝐴, 𝑣} ∈ ran 𝐸 → (𝑣 = 𝐴 → (𝑣 = 𝐵𝑣 = 𝐶))))
5150adantr 480 . . . . . . . . . . . 12 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸)) → ({𝐴, 𝑣} ∈ ran 𝐸 → (𝑣 = 𝐴 → (𝑣 = 𝐵𝑣 = 𝐶))))
5251com3r 85 . . . . . . . . . . 11 (𝑣 = 𝐴 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸)) → ({𝐴, 𝑣} ∈ ran 𝐸 → (𝑣 = 𝐵𝑣 = 𝐶))))
53 orc 399 . . . . . . . . . . . . 13 (𝑣 = 𝐵 → (𝑣 = 𝐵𝑣 = 𝐶))
5453a1d 25 . . . . . . . . . . . 12 (𝑣 = 𝐵 → ({𝐴, 𝑣} ∈ ran 𝐸 → (𝑣 = 𝐵𝑣 = 𝐶)))
5554a1d 25 . . . . . . . . . . 11 (𝑣 = 𝐵 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸)) → ({𝐴, 𝑣} ∈ ran 𝐸 → (𝑣 = 𝐵𝑣 = 𝐶))))
56 olc 398 . . . . . . . . . . . . 13 (𝑣 = 𝐶 → (𝑣 = 𝐵𝑣 = 𝐶))
5756a1d 25 . . . . . . . . . . . 12 (𝑣 = 𝐶 → ({𝐴, 𝑣} ∈ ran 𝐸 → (𝑣 = 𝐵𝑣 = 𝐶)))
5857a1d 25 . . . . . . . . . . 11 (𝑣 = 𝐶 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸)) → ({𝐴, 𝑣} ∈ ran 𝐸 → (𝑣 = 𝐵𝑣 = 𝐶))))
5952, 55, 583jaoi 1383 . . . . . . . . . 10 ((𝑣 = 𝐴𝑣 = 𝐵𝑣 = 𝐶) → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸)) → ({𝐴, 𝑣} ∈ ran 𝐸 → (𝑣 = 𝐵𝑣 = 𝐶))))
6041, 59sylbi 206 . . . . . . . . 9 (𝑣 ∈ {𝐴, 𝐵, 𝐶} → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸)) → ({𝐴, 𝑣} ∈ ran 𝐸 → (𝑣 = 𝐵𝑣 = 𝐶))))
6160com12 32 . . . . . . . 8 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸)) → (𝑣 ∈ {𝐴, 𝐵, 𝐶} → ({𝐴, 𝑣} ∈ ran 𝐸 → (𝑣 = 𝐵𝑣 = 𝐶))))
6239, 61sylbid 229 . . . . . . 7 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸)) → (𝑣𝑉 → ({𝐴, 𝑣} ∈ ran 𝐸 → (𝑣 = 𝐵𝑣 = 𝐶))))
6362impd 446 . . . . . 6 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸)) → ((𝑣𝑉 ∧ {𝐴, 𝑣} ∈ ran 𝐸) → (𝑣 = 𝐵𝑣 = 𝐶)))
64 eqid 2610 . . . . . . . . . . . . . . . . . . . . 21 𝐵 = 𝐵
65643mix2i 1227 . . . . . . . . . . . . . . . . . . . 20 (𝐵 = 𝐴𝐵 = 𝐵𝐵 = 𝐶)
66 eltpg 4174 . . . . . . . . . . . . . . . . . . . 20 (𝐵𝑌 → (𝐵 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝐵 = 𝐴𝐵 = 𝐵𝐵 = 𝐶)))
6765, 66mpbiri 247 . . . . . . . . . . . . . . . . . . 19 (𝐵𝑌𝐵 ∈ {𝐴, 𝐵, 𝐶})
6867a1d 25 . . . . . . . . . . . . . . . . . 18 (𝐵𝑌 → ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸) → 𝐵 ∈ {𝐴, 𝐵, 𝐶}))
69683ad2ant2 1076 . . . . . . . . . . . . . . . . 17 ((𝐴𝑋𝐵𝑌𝐶𝑍) → ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸) → 𝐵 ∈ {𝐴, 𝐵, 𝐶}))
7069imp 444 . . . . . . . . . . . . . . . 16 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) → 𝐵 ∈ {𝐴, 𝐵, 𝐶})
7170adantr 480 . . . . . . . . . . . . . . 15 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ 𝑣 = 𝐵) → 𝐵 ∈ {𝐴, 𝐵, 𝐶})
72 eleq1 2676 . . . . . . . . . . . . . . . . 17 (𝑣 = 𝐵 → (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝐵 ∈ {𝐴, 𝐵, 𝐶}))
7372bicomd 212 . . . . . . . . . . . . . . . 16 (𝑣 = 𝐵 → (𝐵 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣 ∈ {𝐴, 𝐵, 𝐶}))
7473adantl 481 . . . . . . . . . . . . . . 15 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ 𝑣 = 𝐵) → (𝐵 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣 ∈ {𝐴, 𝐵, 𝐶}))
7571, 74mpbid 221 . . . . . . . . . . . . . 14 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ 𝑣 = 𝐵) → 𝑣 ∈ {𝐴, 𝐵, 𝐶})
7637bicomd 212 . . . . . . . . . . . . . . . 16 (𝑉 = {𝐴, 𝐵, 𝐶} → (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣𝑉))
7776ad2antrl 760 . . . . . . . . . . . . . . 15 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) → (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣𝑉))
7877adantr 480 . . . . . . . . . . . . . 14 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ 𝑣 = 𝐵) → (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣𝑉))
7975, 78mpbid 221 . . . . . . . . . . . . 13 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ 𝑣 = 𝐵) → 𝑣𝑉)
8079ex 449 . . . . . . . . . . . 12 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) → (𝑣 = 𝐵𝑣𝑉))
8180adantr 480 . . . . . . . . . . 11 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸)) → (𝑣 = 𝐵𝑣𝑉))
8281impcom 445 . . . . . . . . . 10 ((𝑣 = 𝐵 ∧ (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸))) → 𝑣𝑉)
83 preq2 4213 . . . . . . . . . . . . . . 15 (𝐵 = 𝑣 → {𝐴, 𝐵} = {𝐴, 𝑣})
8483eleq1d 2672 . . . . . . . . . . . . . 14 (𝐵 = 𝑣 → ({𝐴, 𝐵} ∈ ran 𝐸 ↔ {𝐴, 𝑣} ∈ ran 𝐸))
8584eqcoms 2618 . . . . . . . . . . . . 13 (𝑣 = 𝐵 → ({𝐴, 𝐵} ∈ ran 𝐸 ↔ {𝐴, 𝑣} ∈ ran 𝐸))
8685biimpcd 238 . . . . . . . . . . . 12 ({𝐴, 𝐵} ∈ ran 𝐸 → (𝑣 = 𝐵 → {𝐴, 𝑣} ∈ ran 𝐸))
8786ad2antrl 760 . . . . . . . . . . 11 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸)) → (𝑣 = 𝐵 → {𝐴, 𝑣} ∈ ran 𝐸))
8887impcom 445 . . . . . . . . . 10 ((𝑣 = 𝐵 ∧ (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸))) → {𝐴, 𝑣} ∈ ran 𝐸)
8982, 88jca 553 . . . . . . . . 9 ((𝑣 = 𝐵 ∧ (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸))) → (𝑣𝑉 ∧ {𝐴, 𝑣} ∈ ran 𝐸))
9089ex 449 . . . . . . . 8 (𝑣 = 𝐵 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸)) → (𝑣𝑉 ∧ {𝐴, 𝑣} ∈ ran 𝐸)))
91 tpid3g 4248 . . . . . . . . . . . . . . . . . . 19 (𝐶𝑍𝐶 ∈ {𝐴, 𝐵, 𝐶})
9291a1d 25 . . . . . . . . . . . . . . . . . 18 (𝐶𝑍 → ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸) → 𝐶 ∈ {𝐴, 𝐵, 𝐶}))
93923ad2ant3 1077 . . . . . . . . . . . . . . . . 17 ((𝐴𝑋𝐵𝑌𝐶𝑍) → ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸) → 𝐶 ∈ {𝐴, 𝐵, 𝐶}))
9493imp 444 . . . . . . . . . . . . . . . 16 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) → 𝐶 ∈ {𝐴, 𝐵, 𝐶})
9594adantr 480 . . . . . . . . . . . . . . 15 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ 𝑣 = 𝐶) → 𝐶 ∈ {𝐴, 𝐵, 𝐶})
96 eleq1 2676 . . . . . . . . . . . . . . . . 17 (𝑣 = 𝐶 → (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝐶 ∈ {𝐴, 𝐵, 𝐶}))
9796bicomd 212 . . . . . . . . . . . . . . . 16 (𝑣 = 𝐶 → (𝐶 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣 ∈ {𝐴, 𝐵, 𝐶}))
9897adantl 481 . . . . . . . . . . . . . . 15 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ 𝑣 = 𝐶) → (𝐶 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣 ∈ {𝐴, 𝐵, 𝐶}))
9995, 98mpbid 221 . . . . . . . . . . . . . 14 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ 𝑣 = 𝐶) → 𝑣 ∈ {𝐴, 𝐵, 𝐶})
10077adantr 480 . . . . . . . . . . . . . 14 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ 𝑣 = 𝐶) → (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣𝑉))
10199, 100mpbid 221 . . . . . . . . . . . . 13 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ 𝑣 = 𝐶) → 𝑣𝑉)
102101ex 449 . . . . . . . . . . . 12 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) → (𝑣 = 𝐶𝑣𝑉))
103102adantr 480 . . . . . . . . . . 11 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸)) → (𝑣 = 𝐶𝑣𝑉))
104103impcom 445 . . . . . . . . . 10 ((𝑣 = 𝐶 ∧ (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸))) → 𝑣𝑉)
105 preq2 4213 . . . . . . . . . . . . . . 15 (𝐶 = 𝑣 → {𝐴, 𝐶} = {𝐴, 𝑣})
106105eleq1d 2672 . . . . . . . . . . . . . 14 (𝐶 = 𝑣 → ({𝐴, 𝐶} ∈ ran 𝐸 ↔ {𝐴, 𝑣} ∈ ran 𝐸))
107106eqcoms 2618 . . . . . . . . . . . . 13 (𝑣 = 𝐶 → ({𝐴, 𝐶} ∈ ran 𝐸 ↔ {𝐴, 𝑣} ∈ ran 𝐸))
108107biimpcd 238 . . . . . . . . . . . 12 ({𝐴, 𝐶} ∈ ran 𝐸 → (𝑣 = 𝐶 → {𝐴, 𝑣} ∈ ran 𝐸))
109108ad2antll 761 . . . . . . . . . . 11 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸)) → (𝑣 = 𝐶 → {𝐴, 𝑣} ∈ ran 𝐸))
110109impcom 445 . . . . . . . . . 10 ((𝑣 = 𝐶 ∧ (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸))) → {𝐴, 𝑣} ∈ ran 𝐸)
111104, 110jca 553 . . . . . . . . 9 ((𝑣 = 𝐶 ∧ (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸))) → (𝑣𝑉 ∧ {𝐴, 𝑣} ∈ ran 𝐸))
112111ex 449 . . . . . . . 8 (𝑣 = 𝐶 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸)) → (𝑣𝑉 ∧ {𝐴, 𝑣} ∈ ran 𝐸)))
11390, 112jaoi 393 . . . . . . 7 ((𝑣 = 𝐵𝑣 = 𝐶) → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸)) → (𝑣𝑉 ∧ {𝐴, 𝑣} ∈ ran 𝐸)))
114113com12 32 . . . . . 6 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸)) → ((𝑣 = 𝐵𝑣 = 𝐶) → (𝑣𝑉 ∧ {𝐴, 𝑣} ∈ ran 𝐸)))
11563, 114impbid 201 . . . . 5 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸)) → ((𝑣𝑉 ∧ {𝐴, 𝑣} ∈ ran 𝐸) ↔ (𝑣 = 𝐵𝑣 = 𝐶)))
116115abbidv 2728 . . . 4 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸)) → {𝑣 ∣ (𝑣𝑉 ∧ {𝐴, 𝑣} ∈ ran 𝐸)} = {𝑣 ∣ (𝑣 = 𝐵𝑣 = 𝐶)})
117 df-rab 2905 . . . 4 {𝑣𝑉 ∣ {𝐴, 𝑣} ∈ ran 𝐸} = {𝑣 ∣ (𝑣𝑉 ∧ {𝐴, 𝑣} ∈ ran 𝐸)}
118 dfpr2 4143 . . . 4 {𝐵, 𝐶} = {𝑣 ∣ (𝑣 = 𝐵𝑣 = 𝐶)}
119116, 117, 1183eqtr4g 2669 . . 3 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸)) → {𝑣𝑉 ∣ {𝐴, 𝑣} ∈ ran 𝐸} = {𝐵, 𝐶})
12036, 119eqtrd 2644 . 2 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸)) → (⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝐵, 𝐶})
12133, 120impbida 873 1 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) → ((⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝐵, 𝐶} ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 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  ⟨cop 4131   class class class wbr 4583  ran crn 5039  (class class class)co 6549   USGrph cusg 25859   Neighbors cnbgra 25946 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-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-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-usgra 25862  df-nbgra 25949 This theorem is referenced by:  nb3grapr  25982  nb3grapr2  25983  nb3gra2nb  25984
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