Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  1to3vfriswmgr Structured version   Visualization version   GIF version

Theorem 1to3vfriswmgr 41450
 Description: Every friendship graph with one, two or three vertices is a windmill graph. (Contributed by Alexander van der Vekens, 6-Oct-2017.) (Revised by AV, 31-Mar-2021.)
Hypotheses
Ref Expression
3vfriswmgr.v 𝑉 = (Vtx‘𝐺)
3vfriswmgr.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
1to3vfriswmgr ((𝐴𝑋 ∧ (𝑉 = {𝐴} ∨ 𝑉 = {𝐴, 𝐵} ∨ 𝑉 = {𝐴, 𝐵, 𝐶})) → (𝐺 ∈ FriendGraph → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸)))
Distinct variable groups:   𝑤,𝐴   𝑤,𝐵   𝑤,𝐶   𝑤,𝐸   𝑤,𝐺   𝑤,𝑉   𝑤,𝑋   𝐴,,𝑣,𝑤   𝐵,,𝑣   𝐶,,𝑣   ,𝐸,𝑣   ,𝑉,𝑣
Allowed substitution hints:   𝐺(𝑣,)   𝑋(𝑣,)

Proof of Theorem 1to3vfriswmgr
StepHypRef Expression
1 df-3or 1032 . . 3 ((𝑉 = {𝐴} ∨ 𝑉 = {𝐴, 𝐵} ∨ 𝑉 = {𝐴, 𝐵, 𝐶}) ↔ ((𝑉 = {𝐴} ∨ 𝑉 = {𝐴, 𝐵}) ∨ 𝑉 = {𝐴, 𝐵, 𝐶}))
2 3vfriswmgr.v . . . . . 6 𝑉 = (Vtx‘𝐺)
3 3vfriswmgr.e . . . . . 6 𝐸 = (Edg‘𝐺)
42, 31to2vfriswmgr 41449 . . . . 5 ((𝐴𝑋 ∧ (𝑉 = {𝐴} ∨ 𝑉 = {𝐴, 𝐵})) → (𝐺 ∈ FriendGraph → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸)))
54expcom 450 . . . 4 ((𝑉 = {𝐴} ∨ 𝑉 = {𝐴, 𝐵}) → (𝐴𝑋 → (𝐺 ∈ FriendGraph → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸))))
6 tppreq3 4238 . . . . . . 7 (𝐵 = 𝐶 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
76eqeq2d 2620 . . . . . 6 (𝐵 = 𝐶 → (𝑉 = {𝐴, 𝐵, 𝐶} ↔ 𝑉 = {𝐴, 𝐵}))
8 olc 398 . . . . . . . . . 10 (𝑉 = {𝐴, 𝐵} → (𝑉 = {𝐴} ∨ 𝑉 = {𝐴, 𝐵}))
98anim1i 590 . . . . . . . . 9 ((𝑉 = {𝐴, 𝐵} ∧ 𝐴𝑋) → ((𝑉 = {𝐴} ∨ 𝑉 = {𝐴, 𝐵}) ∧ 𝐴𝑋))
109ancomd 466 . . . . . . . 8 ((𝑉 = {𝐴, 𝐵} ∧ 𝐴𝑋) → (𝐴𝑋 ∧ (𝑉 = {𝐴} ∨ 𝑉 = {𝐴, 𝐵})))
1110, 4syl 17 . . . . . . 7 ((𝑉 = {𝐴, 𝐵} ∧ 𝐴𝑋) → (𝐺 ∈ FriendGraph → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸)))
1211ex 449 . . . . . 6 (𝑉 = {𝐴, 𝐵} → (𝐴𝑋 → (𝐺 ∈ FriendGraph → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸))))
137, 12syl6bi 242 . . . . 5 (𝐵 = 𝐶 → (𝑉 = {𝐴, 𝐵, 𝐶} → (𝐴𝑋 → (𝐺 ∈ FriendGraph → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸)))))
14 tpprceq3 4276 . . . . . . . 8 (¬ (𝐵 ∈ V ∧ 𝐵𝐴) → {𝐶, 𝐴, 𝐵} = {𝐶, 𝐴})
15 tprot 4228 . . . . . . . . . . . . 13 {𝐶, 𝐴, 𝐵} = {𝐴, 𝐵, 𝐶}
1615eqeq1i 2615 . . . . . . . . . . . 12 ({𝐶, 𝐴, 𝐵} = {𝐶, 𝐴} ↔ {𝐴, 𝐵, 𝐶} = {𝐶, 𝐴})
1716biimpi 205 . . . . . . . . . . 11 ({𝐶, 𝐴, 𝐵} = {𝐶, 𝐴} → {𝐴, 𝐵, 𝐶} = {𝐶, 𝐴})
18 prcom 4211 . . . . . . . . . . 11 {𝐶, 𝐴} = {𝐴, 𝐶}
1917, 18syl6eq 2660 . . . . . . . . . 10 ({𝐶, 𝐴, 𝐵} = {𝐶, 𝐴} → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐶})
2019eqeq2d 2620 . . . . . . . . 9 ({𝐶, 𝐴, 𝐵} = {𝐶, 𝐴} → (𝑉 = {𝐴, 𝐵, 𝐶} ↔ 𝑉 = {𝐴, 𝐶}))
21 olc 398 . . . . . . . . . . 11 (𝑉 = {𝐴, 𝐶} → (𝑉 = {𝐴} ∨ 𝑉 = {𝐴, 𝐶}))
222, 31to2vfriswmgr 41449 . . . . . . . . . . 11 ((𝐴𝑋 ∧ (𝑉 = {𝐴} ∨ 𝑉 = {𝐴, 𝐶})) → (𝐺 ∈ FriendGraph → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸)))
2321, 22sylan2 490 . . . . . . . . . 10 ((𝐴𝑋𝑉 = {𝐴, 𝐶}) → (𝐺 ∈ FriendGraph → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸)))
2423expcom 450 . . . . . . . . 9 (𝑉 = {𝐴, 𝐶} → (𝐴𝑋 → (𝐺 ∈ FriendGraph → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸))))
2520, 24syl6bi 242 . . . . . . . 8 ({𝐶, 𝐴, 𝐵} = {𝐶, 𝐴} → (𝑉 = {𝐴, 𝐵, 𝐶} → (𝐴𝑋 → (𝐺 ∈ FriendGraph → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸)))))
2614, 25syl 17 . . . . . . 7 (¬ (𝐵 ∈ V ∧ 𝐵𝐴) → (𝑉 = {𝐴, 𝐵, 𝐶} → (𝐴𝑋 → (𝐺 ∈ FriendGraph → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸)))))
2726a1d 25 . . . . . 6 (¬ (𝐵 ∈ V ∧ 𝐵𝐴) → (𝐵𝐶 → (𝑉 = {𝐴, 𝐵, 𝐶} → (𝐴𝑋 → (𝐺 ∈ FriendGraph → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸))))))
28 tpprceq3 4276 . . . . . . . 8 (¬ (𝐶 ∈ V ∧ 𝐶𝐴) → {𝐵, 𝐴, 𝐶} = {𝐵, 𝐴})
29 tpcoma 4229 . . . . . . . . . . . . 13 {𝐵, 𝐴, 𝐶} = {𝐴, 𝐵, 𝐶}
3029eqeq1i 2615 . . . . . . . . . . . 12 ({𝐵, 𝐴, 𝐶} = {𝐵, 𝐴} ↔ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐴})
3130biimpi 205 . . . . . . . . . . 11 ({𝐵, 𝐴, 𝐶} = {𝐵, 𝐴} → {𝐴, 𝐵, 𝐶} = {𝐵, 𝐴})
32 prcom 4211 . . . . . . . . . . 11 {𝐵, 𝐴} = {𝐴, 𝐵}
3331, 32syl6eq 2660 . . . . . . . . . 10 ({𝐵, 𝐴, 𝐶} = {𝐵, 𝐴} → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
3433eqeq2d 2620 . . . . . . . . 9 ({𝐵, 𝐴, 𝐶} = {𝐵, 𝐴} → (𝑉 = {𝐴, 𝐵, 𝐶} ↔ 𝑉 = {𝐴, 𝐵}))
358, 4sylan2 490 . . . . . . . . . . 11 ((𝐴𝑋𝑉 = {𝐴, 𝐵}) → (𝐺 ∈ FriendGraph → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸)))
3635expcom 450 . . . . . . . . . 10 (𝑉 = {𝐴, 𝐵} → (𝐴𝑋 → (𝐺 ∈ FriendGraph → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸))))
3736a1d 25 . . . . . . . . 9 (𝑉 = {𝐴, 𝐵} → (𝐵𝐶 → (𝐴𝑋 → (𝐺 ∈ FriendGraph → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸)))))
3834, 37syl6bi 242 . . . . . . . 8 ({𝐵, 𝐴, 𝐶} = {𝐵, 𝐴} → (𝑉 = {𝐴, 𝐵, 𝐶} → (𝐵𝐶 → (𝐴𝑋 → (𝐺 ∈ FriendGraph → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸))))))
3928, 38syl 17 . . . . . . 7 (¬ (𝐶 ∈ V ∧ 𝐶𝐴) → (𝑉 = {𝐴, 𝐵, 𝐶} → (𝐵𝐶 → (𝐴𝑋 → (𝐺 ∈ FriendGraph → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸))))))
4039com23 84 . . . . . 6 (¬ (𝐶 ∈ V ∧ 𝐶𝐴) → (𝐵𝐶 → (𝑉 = {𝐴, 𝐵, 𝐶} → (𝐴𝑋 → (𝐺 ∈ FriendGraph → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸))))))
41 simpl 472 . . . . . . . . . . . . 13 ((𝐵 ∈ V ∧ 𝐵𝐴) → 𝐵 ∈ V)
42 simpl 472 . . . . . . . . . . . . 13 ((𝐶 ∈ V ∧ 𝐶𝐴) → 𝐶 ∈ V)
4341, 42anim12i 588 . . . . . . . . . . . 12 (((𝐵 ∈ V ∧ 𝐵𝐴) ∧ (𝐶 ∈ V ∧ 𝐶𝐴)) → (𝐵 ∈ V ∧ 𝐶 ∈ V))
4443ad2antrr 758 . . . . . . . . . . 11 (((((𝐵 ∈ V ∧ 𝐵𝐴) ∧ (𝐶 ∈ V ∧ 𝐶𝐴)) ∧ 𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → (𝐵 ∈ V ∧ 𝐶 ∈ V))
4544anim1i 590 . . . . . . . . . 10 ((((((𝐵 ∈ V ∧ 𝐵𝐴) ∧ (𝐶 ∈ V ∧ 𝐶𝐴)) ∧ 𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐴𝑋) → ((𝐵 ∈ V ∧ 𝐶 ∈ V) ∧ 𝐴𝑋))
4645ancomd 466 . . . . . . . . 9 ((((((𝐵 ∈ V ∧ 𝐵𝐴) ∧ (𝐶 ∈ V ∧ 𝐶𝐴)) ∧ 𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐴𝑋) → (𝐴𝑋 ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)))
47 3anass 1035 . . . . . . . . 9 ((𝐴𝑋𝐵 ∈ V ∧ 𝐶 ∈ V) ↔ (𝐴𝑋 ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)))
4846, 47sylibr 223 . . . . . . . 8 ((((((𝐵 ∈ V ∧ 𝐵𝐴) ∧ (𝐶 ∈ V ∧ 𝐶𝐴)) ∧ 𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐴𝑋) → (𝐴𝑋𝐵 ∈ V ∧ 𝐶 ∈ V))
49 simpr 476 . . . . . . . . . . . . 13 ((𝐵 ∈ V ∧ 𝐵𝐴) → 𝐵𝐴)
5049necomd 2837 . . . . . . . . . . . 12 ((𝐵 ∈ V ∧ 𝐵𝐴) → 𝐴𝐵)
51 simpr 476 . . . . . . . . . . . . 13 ((𝐶 ∈ V ∧ 𝐶𝐴) → 𝐶𝐴)
5251necomd 2837 . . . . . . . . . . . 12 ((𝐶 ∈ V ∧ 𝐶𝐴) → 𝐴𝐶)
5350, 52anim12i 588 . . . . . . . . . . 11 (((𝐵 ∈ V ∧ 𝐵𝐴) ∧ (𝐶 ∈ V ∧ 𝐶𝐴)) → (𝐴𝐵𝐴𝐶))
5453anim1i 590 . . . . . . . . . 10 ((((𝐵 ∈ V ∧ 𝐵𝐴) ∧ (𝐶 ∈ V ∧ 𝐶𝐴)) ∧ 𝐵𝐶) → ((𝐴𝐵𝐴𝐶) ∧ 𝐵𝐶))
55 df-3an 1033 . . . . . . . . . 10 ((𝐴𝐵𝐴𝐶𝐵𝐶) ↔ ((𝐴𝐵𝐴𝐶) ∧ 𝐵𝐶))
5654, 55sylibr 223 . . . . . . . . 9 ((((𝐵 ∈ V ∧ 𝐵𝐴) ∧ (𝐶 ∈ V ∧ 𝐶𝐴)) ∧ 𝐵𝐶) → (𝐴𝐵𝐴𝐶𝐵𝐶))
5756ad2antrr 758 . . . . . . . 8 ((((((𝐵 ∈ V ∧ 𝐵𝐴) ∧ (𝐶 ∈ V ∧ 𝐶𝐴)) ∧ 𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐴𝑋) → (𝐴𝐵𝐴𝐶𝐵𝐶))
58 simplr 788 . . . . . . . 8 ((((((𝐵 ∈ V ∧ 𝐵𝐴) ∧ (𝐶 ∈ V ∧ 𝐶𝐴)) ∧ 𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐴𝑋) → 𝑉 = {𝐴, 𝐵, 𝐶})
592, 33vfriswmgr 41448 . . . . . . . 8 (((𝐴𝑋𝐵 ∈ V ∧ 𝐶 ∈ V) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → (𝐺 ∈ FriendGraph → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸)))
6048, 57, 58, 59syl3anc 1318 . . . . . . 7 ((((((𝐵 ∈ V ∧ 𝐵𝐴) ∧ (𝐶 ∈ V ∧ 𝐶𝐴)) ∧ 𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐴𝑋) → (𝐺 ∈ FriendGraph → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸)))
6160exp41 636 . . . . . 6 (((𝐵 ∈ V ∧ 𝐵𝐴) ∧ (𝐶 ∈ V ∧ 𝐶𝐴)) → (𝐵𝐶 → (𝑉 = {𝐴, 𝐵, 𝐶} → (𝐴𝑋 → (𝐺 ∈ FriendGraph → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸))))))
6227, 40, 61ecase 980 . . . . 5 (𝐵𝐶 → (𝑉 = {𝐴, 𝐵, 𝐶} → (𝐴𝑋 → (𝐺 ∈ FriendGraph → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸)))))
6313, 62pm2.61ine 2865 . . . 4 (𝑉 = {𝐴, 𝐵, 𝐶} → (𝐴𝑋 → (𝐺 ∈ FriendGraph → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸))))
645, 63jaoi 393 . . 3 (((𝑉 = {𝐴} ∨ 𝑉 = {𝐴, 𝐵}) ∨ 𝑉 = {𝐴, 𝐵, 𝐶}) → (𝐴𝑋 → (𝐺 ∈ FriendGraph → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸))))
651, 64sylbi 206 . 2 ((𝑉 = {𝐴} ∨ 𝑉 = {𝐴, 𝐵} ∨ 𝑉 = {𝐴, 𝐵, 𝐶}) → (𝐴𝑋 → (𝐺 ∈ FriendGraph → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸))))
6665impcom 445 1 ((𝐴𝑋 ∧ (𝑉 = {𝐴} ∨ 𝑉 = {𝐴, 𝐵} ∨ 𝑉 = {𝐴, 𝐵, 𝐶})) → (𝐺 ∈ FriendGraph → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   ∧ wa 383   ∨ w3o 1030   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  ∃!wreu 2898  Vcvv 3173   ∖ cdif 3537  {csn 4125  {cpr 4127  {ctp 4129  ‘cfv 5804  Vtxcvtx 25673  Edgcedga 25792   FriendGraph cfrgr 41428 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-umgr 25750  df-edga 25793  df-usgr 40381  df-frgr 41429 This theorem is referenced by:  1to3vfriendship-av  41451
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