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Theorem 1to3vfriswmgr 27629
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 1109 . . 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 27628 . . . . 5 ((𝐴𝑋 ∧ (𝑉 = {𝐴} ∨ 𝑉 = {𝐴, 𝐵})) → (𝐺 ∈ FriendGraph → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸)))
54expcom 403 . . . 4 ((𝑉 = {𝐴} ∨ 𝑉 = {𝐴, 𝐵}) → (𝐴𝑋 → (𝐺 ∈ FriendGraph → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸))))
6 tppreq3 4483 . . . . . . 7 (𝐵 = 𝐶 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
76eqeq2d 2809 . . . . . 6 (𝐵 = 𝐶 → (𝑉 = {𝐴, 𝐵, 𝐶} ↔ 𝑉 = {𝐴, 𝐵}))
8 olc 895 . . . . . . . . . 10 (𝑉 = {𝐴, 𝐵} → (𝑉 = {𝐴} ∨ 𝑉 = {𝐴, 𝐵}))
98anim1i 609 . . . . . . . . 9 ((𝑉 = {𝐴, 𝐵} ∧ 𝐴𝑋) → ((𝑉 = {𝐴} ∨ 𝑉 = {𝐴, 𝐵}) ∧ 𝐴𝑋))
109ancomd 454 . . . . . . . 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 402 . . . . . 6 (𝑉 = {𝐴, 𝐵} → (𝐴𝑋 → (𝐺 ∈ FriendGraph → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸))))
137, 12syl6bi 245 . . . . 5 (𝐵 = 𝐶 → (𝑉 = {𝐴, 𝐵, 𝐶} → (𝐴𝑋 → (𝐺 ∈ FriendGraph → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸)))))
14 tpprceq3 4523 . . . . . . . 8 (¬ (𝐵 ∈ V ∧ 𝐵𝐴) → {𝐶, 𝐴, 𝐵} = {𝐶, 𝐴})
15 tprot 4473 . . . . . . . . . . . . 13 {𝐶, 𝐴, 𝐵} = {𝐴, 𝐵, 𝐶}
1615eqeq1i 2804 . . . . . . . . . . . 12 ({𝐶, 𝐴, 𝐵} = {𝐶, 𝐴} ↔ {𝐴, 𝐵, 𝐶} = {𝐶, 𝐴})
1716biimpi 208 . . . . . . . . . . 11 ({𝐶, 𝐴, 𝐵} = {𝐶, 𝐴} → {𝐴, 𝐵, 𝐶} = {𝐶, 𝐴})
18 prcom 4456 . . . . . . . . . . 11 {𝐶, 𝐴} = {𝐴, 𝐶}
1917, 18syl6eq 2849 . . . . . . . . . 10 ({𝐶, 𝐴, 𝐵} = {𝐶, 𝐴} → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐶})
2019eqeq2d 2809 . . . . . . . . 9 ({𝐶, 𝐴, 𝐵} = {𝐶, 𝐴} → (𝑉 = {𝐴, 𝐵, 𝐶} ↔ 𝑉 = {𝐴, 𝐶}))
21 olc 895 . . . . . . . . . . 11 (𝑉 = {𝐴, 𝐶} → (𝑉 = {𝐴} ∨ 𝑉 = {𝐴, 𝐶}))
222, 31to2vfriswmgr 27628 . . . . . . . . . . 11 ((𝐴𝑋 ∧ (𝑉 = {𝐴} ∨ 𝑉 = {𝐴, 𝐶})) → (𝐺 ∈ FriendGraph → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸)))
2321, 22sylan2 587 . . . . . . . . . 10 ((𝐴𝑋𝑉 = {𝐴, 𝐶}) → (𝐺 ∈ FriendGraph → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸)))
2423expcom 403 . . . . . . . . 9 (𝑉 = {𝐴, 𝐶} → (𝐴𝑋 → (𝐺 ∈ FriendGraph → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸))))
2520, 24syl6bi 245 . . . . . . . 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 4523 . . . . . . . 8 (¬ (𝐶 ∈ V ∧ 𝐶𝐴) → {𝐵, 𝐴, 𝐶} = {𝐵, 𝐴})
29 tpcoma 4474 . . . . . . . . . . . . 13 {𝐵, 𝐴, 𝐶} = {𝐴, 𝐵, 𝐶}
3029eqeq1i 2804 . . . . . . . . . . . 12 ({𝐵, 𝐴, 𝐶} = {𝐵, 𝐴} ↔ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐴})
3130biimpi 208 . . . . . . . . . . 11 ({𝐵, 𝐴, 𝐶} = {𝐵, 𝐴} → {𝐴, 𝐵, 𝐶} = {𝐵, 𝐴})
32 prcom 4456 . . . . . . . . . . 11 {𝐵, 𝐴} = {𝐴, 𝐵}
3331, 32syl6eq 2849 . . . . . . . . . 10 ({𝐵, 𝐴, 𝐶} = {𝐵, 𝐴} → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
3433eqeq2d 2809 . . . . . . . . 9 ({𝐵, 𝐴, 𝐶} = {𝐵, 𝐴} → (𝑉 = {𝐴, 𝐵, 𝐶} ↔ 𝑉 = {𝐴, 𝐵}))
358, 4sylan2 587 . . . . . . . . . . 11 ((𝐴𝑋𝑉 = {𝐴, 𝐵}) → (𝐺 ∈ FriendGraph → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸)))
3635expcom 403 . . . . . . . . . 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 245 . . . . . . . 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 475 . . . . . . . . . . . . 13 ((𝐵 ∈ V ∧ 𝐵𝐴) → 𝐵 ∈ V)
42 simpl 475 . . . . . . . . . . . . 13 ((𝐶 ∈ V ∧ 𝐶𝐴) → 𝐶 ∈ V)
4341, 42anim12i 607 . . . . . . . . . . . 12 (((𝐵 ∈ V ∧ 𝐵𝐴) ∧ (𝐶 ∈ V ∧ 𝐶𝐴)) → (𝐵 ∈ V ∧ 𝐶 ∈ V))
4443ad2antrr 718 . . . . . . . . . . 11 (((((𝐵 ∈ V ∧ 𝐵𝐴) ∧ (𝐶 ∈ V ∧ 𝐶𝐴)) ∧ 𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → (𝐵 ∈ V ∧ 𝐶 ∈ V))
4544anim1i 609 . . . . . . . . . 10 ((((((𝐵 ∈ V ∧ 𝐵𝐴) ∧ (𝐶 ∈ V ∧ 𝐶𝐴)) ∧ 𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐴𝑋) → ((𝐵 ∈ V ∧ 𝐶 ∈ V) ∧ 𝐴𝑋))
4645ancomd 454 . . . . . . . . 9 ((((((𝐵 ∈ V ∧ 𝐵𝐴) ∧ (𝐶 ∈ V ∧ 𝐶𝐴)) ∧ 𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐴𝑋) → (𝐴𝑋 ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)))
47 3anass 1117 . . . . . . . . 9 ((𝐴𝑋𝐵 ∈ V ∧ 𝐶 ∈ V) ↔ (𝐴𝑋 ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)))
4846, 47sylibr 226 . . . . . . . 8 ((((((𝐵 ∈ V ∧ 𝐵𝐴) ∧ (𝐶 ∈ V ∧ 𝐶𝐴)) ∧ 𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐴𝑋) → (𝐴𝑋𝐵 ∈ V ∧ 𝐶 ∈ V))
49 simpr 478 . . . . . . . . . . . . 13 ((𝐵 ∈ V ∧ 𝐵𝐴) → 𝐵𝐴)
5049necomd 3026 . . . . . . . . . . . 12 ((𝐵 ∈ V ∧ 𝐵𝐴) → 𝐴𝐵)
51 simpr 478 . . . . . . . . . . . . 13 ((𝐶 ∈ V ∧ 𝐶𝐴) → 𝐶𝐴)
5251necomd 3026 . . . . . . . . . . . 12 ((𝐶 ∈ V ∧ 𝐶𝐴) → 𝐴𝐶)
5350, 52anim12i 607 . . . . . . . . . . 11 (((𝐵 ∈ V ∧ 𝐵𝐴) ∧ (𝐶 ∈ V ∧ 𝐶𝐴)) → (𝐴𝐵𝐴𝐶))
5453anim1i 609 . . . . . . . . . 10 ((((𝐵 ∈ V ∧ 𝐵𝐴) ∧ (𝐶 ∈ V ∧ 𝐶𝐴)) ∧ 𝐵𝐶) → ((𝐴𝐵𝐴𝐶) ∧ 𝐵𝐶))
55 df-3an 1110 . . . . . . . . . 10 ((𝐴𝐵𝐴𝐶𝐵𝐶) ↔ ((𝐴𝐵𝐴𝐶) ∧ 𝐵𝐶))
5654, 55sylibr 226 . . . . . . . . 9 ((((𝐵 ∈ V ∧ 𝐵𝐴) ∧ (𝐶 ∈ V ∧ 𝐶𝐴)) ∧ 𝐵𝐶) → (𝐴𝐵𝐴𝐶𝐵𝐶))
5756ad2antrr 718 . . . . . . . 8 ((((((𝐵 ∈ V ∧ 𝐵𝐴) ∧ (𝐶 ∈ V ∧ 𝐶𝐴)) ∧ 𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐴𝑋) → (𝐴𝐵𝐴𝐶𝐵𝐶))
58 simplr 786 . . . . . . . 8 ((((((𝐵 ∈ V ∧ 𝐵𝐴) ∧ (𝐶 ∈ V ∧ 𝐶𝐴)) ∧ 𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐴𝑋) → 𝑉 = {𝐴, 𝐵, 𝐶})
592, 33vfriswmgr 27627 . . . . . . . 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 1491 . . . . . . 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 426 . . . . . 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 1057 . . . . 5 (𝐵𝐶 → (𝑉 = {𝐴, 𝐵, 𝐶} → (𝐴𝑋 → (𝐺 ∈ FriendGraph → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸)))))
6313, 62pm2.61ine 3054 . . . 4 (𝑉 = {𝐴, 𝐵, 𝐶} → (𝐴𝑋 → (𝐺 ∈ FriendGraph → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸))))
645, 63jaoi 884 . . 3 (((𝑉 = {𝐴} ∨ 𝑉 = {𝐴, 𝐵}) ∨ 𝑉 = {𝐴, 𝐵, 𝐶}) → (𝐴𝑋 → (𝐺 ∈ FriendGraph → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸))))
651, 64sylbi 209 . 2 ((𝑉 = {𝐴} ∨ 𝑉 = {𝐴, 𝐵} ∨ 𝑉 = {𝐴, 𝐵, 𝐶}) → (𝐴𝑋 → (𝐺 ∈ FriendGraph → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸))))
6665impcom 397 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  wa 385  wo 874  w3o 1107  w3a 1108   = wceq 1653  wcel 2157  wne 2971  wral 3089  wrex 3090  ∃!wreu 3091  Vcvv 3385  cdif 3766  {csn 4368  {cpr 4370  {ctp 4372  cfv 6101  Vtxcvtx 26231  Edgcedg 26282   FriendGraph cfrgr 27605
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-cnex 10280  ax-resscn 10281  ax-1cn 10282  ax-icn 10283  ax-addcl 10284  ax-addrcl 10285  ax-mulcl 10286  ax-mulrcl 10287  ax-mulcom 10288  ax-addass 10289  ax-mulass 10290  ax-distr 10291  ax-i2m1 10292  ax-1ne0 10293  ax-1rid 10294  ax-rnegex 10295  ax-rrecex 10296  ax-cnre 10297  ax-pre-lttri 10298  ax-pre-lttrn 10299  ax-pre-ltadd 10300  ax-pre-mulgt0 10301
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-int 4668  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6839  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-om 7300  df-1st 7401  df-2nd 7402  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-1o 7799  df-oadd 7803  df-er 7982  df-en 8196  df-dom 8197  df-sdom 8198  df-fin 8199  df-card 9051  df-cda 9278  df-pnf 10365  df-mnf 10366  df-xr 10367  df-ltxr 10368  df-le 10369  df-sub 10558  df-neg 10559  df-nn 11313  df-2 11376  df-n0 11581  df-z 11667  df-uz 11931  df-fz 12581  df-hash 13371  df-edg 26283  df-umgr 26318  df-usgr 26387  df-frgr 27606
This theorem is referenced by:  1to3vfriendship  27630
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