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Theorem 1to3vfriswmgr 27405
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 1073 . . 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 27404 . . . . 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 4426 . . . . . . 7 (𝐵 = 𝐶 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
76eqeq2d 2758 . . . . . 6 (𝐵 = 𝐶 → (𝑉 = {𝐴, 𝐵, 𝐶} ↔ 𝑉 = {𝐴, 𝐵}))
8 olc 398 . . . . . . . . . 10 (𝑉 = {𝐴, 𝐵} → (𝑉 = {𝐴} ∨ 𝑉 = {𝐴, 𝐵}))
98anim1i 593 . . . . . . . . 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 243 . . . . 5 (𝐵 = 𝐶 → (𝑉 = {𝐴, 𝐵, 𝐶} → (𝐴𝑋 → (𝐺 ∈ FriendGraph → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸)))))
14 tpprceq3 4468 . . . . . . . 8 (¬ (𝐵 ∈ V ∧ 𝐵𝐴) → {𝐶, 𝐴, 𝐵} = {𝐶, 𝐴})
15 tprot 4416 . . . . . . . . . . . . 13 {𝐶, 𝐴, 𝐵} = {𝐴, 𝐵, 𝐶}
1615eqeq1i 2753 . . . . . . . . . . . 12 ({𝐶, 𝐴, 𝐵} = {𝐶, 𝐴} ↔ {𝐴, 𝐵, 𝐶} = {𝐶, 𝐴})
1716biimpi 206 . . . . . . . . . . 11 ({𝐶, 𝐴, 𝐵} = {𝐶, 𝐴} → {𝐴, 𝐵, 𝐶} = {𝐶, 𝐴})
18 prcom 4399 . . . . . . . . . . 11 {𝐶, 𝐴} = {𝐴, 𝐶}
1917, 18syl6eq 2798 . . . . . . . . . 10 ({𝐶, 𝐴, 𝐵} = {𝐶, 𝐴} → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐶})
2019eqeq2d 2758 . . . . . . . . 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 27404 . . . . . . . . . . 11 ((𝐴𝑋 ∧ (𝑉 = {𝐴} ∨ 𝑉 = {𝐴, 𝐶})) → (𝐺 ∈ FriendGraph → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸)))
2321, 22sylan2 492 . . . . . . . . . 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 243 . . . . . . . 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 4468 . . . . . . . 8 (¬ (𝐶 ∈ V ∧ 𝐶𝐴) → {𝐵, 𝐴, 𝐶} = {𝐵, 𝐴})
29 tpcoma 4417 . . . . . . . . . . . . 13 {𝐵, 𝐴, 𝐶} = {𝐴, 𝐵, 𝐶}
3029eqeq1i 2753 . . . . . . . . . . . 12 ({𝐵, 𝐴, 𝐶} = {𝐵, 𝐴} ↔ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐴})
3130biimpi 206 . . . . . . . . . . 11 ({𝐵, 𝐴, 𝐶} = {𝐵, 𝐴} → {𝐴, 𝐵, 𝐶} = {𝐵, 𝐴})
32 prcom 4399 . . . . . . . . . . 11 {𝐵, 𝐴} = {𝐴, 𝐵}
3331, 32syl6eq 2798 . . . . . . . . . 10 ({𝐵, 𝐴, 𝐶} = {𝐵, 𝐴} → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
3433eqeq2d 2758 . . . . . . . . 9 ({𝐵, 𝐴, 𝐶} = {𝐵, 𝐴} → (𝑉 = {𝐴, 𝐵, 𝐶} ↔ 𝑉 = {𝐴, 𝐵}))
358, 4sylan2 492 . . . . . . . . . . 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 243 . . . . . . . 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 86 . . . . . 6 (¬ (𝐶 ∈ V ∧ 𝐶𝐴) → (𝐵𝐶 → (𝑉 = {𝐴, 𝐵, 𝐶} → (𝐴𝑋 → (𝐺 ∈ FriendGraph → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸))))))
41 simpl 474 . . . . . . . . . . . . 13 ((𝐵 ∈ V ∧ 𝐵𝐴) → 𝐵 ∈ V)
42 simpl 474 . . . . . . . . . . . . 13 ((𝐶 ∈ V ∧ 𝐶𝐴) → 𝐶 ∈ V)
4341, 42anim12i 591 . . . . . . . . . . . 12 (((𝐵 ∈ V ∧ 𝐵𝐴) ∧ (𝐶 ∈ V ∧ 𝐶𝐴)) → (𝐵 ∈ V ∧ 𝐶 ∈ V))
4443ad2antrr 764 . . . . . . . . . . 11 (((((𝐵 ∈ V ∧ 𝐵𝐴) ∧ (𝐶 ∈ V ∧ 𝐶𝐴)) ∧ 𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → (𝐵 ∈ V ∧ 𝐶 ∈ V))
4544anim1i 593 . . . . . . . . . 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 1081 . . . . . . . . 9 ((𝐴𝑋𝐵 ∈ V ∧ 𝐶 ∈ V) ↔ (𝐴𝑋 ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)))
4846, 47sylibr 224 . . . . . . . 8 ((((((𝐵 ∈ V ∧ 𝐵𝐴) ∧ (𝐶 ∈ V ∧ 𝐶𝐴)) ∧ 𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐴𝑋) → (𝐴𝑋𝐵 ∈ V ∧ 𝐶 ∈ V))
49 simpr 479 . . . . . . . . . . . . 13 ((𝐵 ∈ V ∧ 𝐵𝐴) → 𝐵𝐴)
5049necomd 2975 . . . . . . . . . . . 12 ((𝐵 ∈ V ∧ 𝐵𝐴) → 𝐴𝐵)
51 simpr 479 . . . . . . . . . . . . 13 ((𝐶 ∈ V ∧ 𝐶𝐴) → 𝐶𝐴)
5251necomd 2975 . . . . . . . . . . . 12 ((𝐶 ∈ V ∧ 𝐶𝐴) → 𝐴𝐶)
5350, 52anim12i 591 . . . . . . . . . . 11 (((𝐵 ∈ V ∧ 𝐵𝐴) ∧ (𝐶 ∈ V ∧ 𝐶𝐴)) → (𝐴𝐵𝐴𝐶))
5453anim1i 593 . . . . . . . . . 10 ((((𝐵 ∈ V ∧ 𝐵𝐴) ∧ (𝐶 ∈ V ∧ 𝐶𝐴)) ∧ 𝐵𝐶) → ((𝐴𝐵𝐴𝐶) ∧ 𝐵𝐶))
55 df-3an 1074 . . . . . . . . . 10 ((𝐴𝐵𝐴𝐶𝐵𝐶) ↔ ((𝐴𝐵𝐴𝐶) ∧ 𝐵𝐶))
5654, 55sylibr 224 . . . . . . . . 9 ((((𝐵 ∈ V ∧ 𝐵𝐴) ∧ (𝐶 ∈ V ∧ 𝐶𝐴)) ∧ 𝐵𝐶) → (𝐴𝐵𝐴𝐶𝐵𝐶))
5756ad2antrr 764 . . . . . . . 8 ((((((𝐵 ∈ V ∧ 𝐵𝐴) ∧ (𝐶 ∈ V ∧ 𝐶𝐴)) ∧ 𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐴𝑋) → (𝐴𝐵𝐴𝐶𝐵𝐶))
58 simplr 809 . . . . . . . 8 ((((((𝐵 ∈ V ∧ 𝐵𝐴) ∧ (𝐶 ∈ V ∧ 𝐶𝐴)) ∧ 𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐴𝑋) → 𝑉 = {𝐴, 𝐵, 𝐶})
592, 33vfriswmgr 27403 . . . . . . . 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 1463 . . . . . . 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 639 . . . . . 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 1020 . . . . 5 (𝐵𝐶 → (𝑉 = {𝐴, 𝐵, 𝐶} → (𝐴𝑋 → (𝐺 ∈ FriendGraph → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸)))))
6313, 62pm2.61ine 3003 . . . 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 207 . 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 1071  w3a 1072   = wceq 1620  wcel 2127  wne 2920  wral 3038  wrex 3039  ∃!wreu 3040  Vcvv 3328  cdif 3700  {csn 4309  {cpr 4311  {ctp 4313  cfv 6037  Vtxcvtx 26044  Edgcedg 26109   FriendGraph cfrgr 27381
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-rep 4911  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102  ax-cnex 10155  ax-resscn 10156  ax-1cn 10157  ax-icn 10158  ax-addcl 10159  ax-addrcl 10160  ax-mulcl 10161  ax-mulrcl 10162  ax-mulcom 10163  ax-addass 10164  ax-mulass 10165  ax-distr 10166  ax-i2m1 10167  ax-1ne0 10168  ax-1rid 10169  ax-rnegex 10170  ax-rrecex 10171  ax-cnre 10172  ax-pre-lttri 10173  ax-pre-lttrn 10174  ax-pre-ltadd 10175  ax-pre-mulgt0 10176
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-nel 3024  df-ral 3043  df-rex 3044  df-reu 3045  df-rmo 3046  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-pss 3719  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-tp 4314  df-op 4316  df-uni 4577  df-int 4616  df-iun 4662  df-br 4793  df-opab 4853  df-mpt 4870  df-tr 4893  df-id 5162  df-eprel 5167  df-po 5175  df-so 5176  df-fr 5213  df-we 5215  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-pred 5829  df-ord 5875  df-on 5876  df-lim 5877  df-suc 5878  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-f1 6042  df-fo 6043  df-f1o 6044  df-fv 6045  df-riota 6762  df-ov 6804  df-oprab 6805  df-mpt2 6806  df-om 7219  df-1st 7321  df-2nd 7322  df-wrecs 7564  df-recs 7625  df-rdg 7663  df-1o 7717  df-oadd 7721  df-er 7899  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-card 8926  df-cda 9153  df-pnf 10239  df-mnf 10240  df-xr 10241  df-ltxr 10242  df-le 10243  df-sub 10431  df-neg 10432  df-nn 11184  df-2 11242  df-n0 11456  df-z 11541  df-uz 11851  df-fz 12491  df-hash 13283  df-edg 26110  df-umgr 26148  df-usgr 26216  df-frgr 27382
This theorem is referenced by:  1to3vfriendship  27406
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