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Theorem frgra3v 26529
 Description: Any graph with three vertices which are completely connected with each other is a friendship graph. (Contributed by Alexander van der Vekens, 5-Oct-2017.)
Assertion
Ref Expression
frgra3v (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ({𝐴, 𝐵, 𝐶} USGrph 𝐸 → ({𝐴, 𝐵, 𝐶} FriendGrph 𝐸 ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))))

Proof of Theorem frgra3v
Dummy variables 𝑘 𝑙 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 usgrav 25867 . . . . . 6 ({𝐴, 𝐵, 𝐶} USGrph 𝐸 → ({𝐴, 𝐵, 𝐶} ∈ V ∧ 𝐸 ∈ V))
2 isfrgra 26517 . . . . . 6 (({𝐴, 𝐵, 𝐶} ∈ V ∧ 𝐸 ∈ V) → ({𝐴, 𝐵, 𝐶} FriendGrph 𝐸 ↔ ({𝐴, 𝐵, 𝐶} USGrph 𝐸 ∧ ∀𝑘 ∈ {𝐴, 𝐵, 𝐶}∀𝑙 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸)))
31, 2syl 17 . . . . 5 ({𝐴, 𝐵, 𝐶} USGrph 𝐸 → ({𝐴, 𝐵, 𝐶} FriendGrph 𝐸 ↔ ({𝐴, 𝐵, 𝐶} USGrph 𝐸 ∧ ∀𝑘 ∈ {𝐴, 𝐵, 𝐶}∀𝑙 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸)))
43adantl 481 . . . 4 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → ({𝐴, 𝐵, 𝐶} FriendGrph 𝐸 ↔ ({𝐴, 𝐵, 𝐶} USGrph 𝐸 ∧ ∀𝑘 ∈ {𝐴, 𝐵, 𝐶}∀𝑙 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸)))
5 ibar 524 . . . . 5 ({𝐴, 𝐵, 𝐶} USGrph 𝐸 → (∀𝑘 ∈ {𝐴, 𝐵, 𝐶}∀𝑙 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ({𝐴, 𝐵, 𝐶} USGrph 𝐸 ∧ ∀𝑘 ∈ {𝐴, 𝐵, 𝐶}∀𝑙 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸)))
65adantl 481 . . . 4 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → (∀𝑘 ∈ {𝐴, 𝐵, 𝐶}∀𝑙 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ({𝐴, 𝐵, 𝐶} USGrph 𝐸 ∧ ∀𝑘 ∈ {𝐴, 𝐵, 𝐶}∀𝑙 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸)))
74, 6bitr4d 270 . . 3 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → ({𝐴, 𝐵, 𝐶} FriendGrph 𝐸 ↔ ∀𝑘 ∈ {𝐴, 𝐵, 𝐶}∀𝑙 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸))
8 sneq 4135 . . . . . . . . 9 (𝑘 = 𝐴 → {𝑘} = {𝐴})
98difeq2d 3690 . . . . . . . 8 (𝑘 = 𝐴 → ({𝐴, 𝐵, 𝐶} ∖ {𝑘}) = ({𝐴, 𝐵, 𝐶} ∖ {𝐴}))
10 preq2 4213 . . . . . . . . . . 11 (𝑘 = 𝐴 → {𝑥, 𝑘} = {𝑥, 𝐴})
1110preq1d 4218 . . . . . . . . . 10 (𝑘 = 𝐴 → {{𝑥, 𝑘}, {𝑥, 𝑙}} = {{𝑥, 𝐴}, {𝑥, 𝑙}})
1211sseq1d 3595 . . . . . . . . 9 (𝑘 = 𝐴 → ({{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸))
1312reubidv 3103 . . . . . . . 8 (𝑘 = 𝐴 → (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸))
149, 13raleqbidv 3129 . . . . . . 7 (𝑘 = 𝐴 → (∀𝑙 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ∀𝑙 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴})∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸))
15 sneq 4135 . . . . . . . . 9 (𝑘 = 𝐵 → {𝑘} = {𝐵})
1615difeq2d 3690 . . . . . . . 8 (𝑘 = 𝐵 → ({𝐴, 𝐵, 𝐶} ∖ {𝑘}) = ({𝐴, 𝐵, 𝐶} ∖ {𝐵}))
17 preq2 4213 . . . . . . . . . . 11 (𝑘 = 𝐵 → {𝑥, 𝑘} = {𝑥, 𝐵})
1817preq1d 4218 . . . . . . . . . 10 (𝑘 = 𝐵 → {{𝑥, 𝑘}, {𝑥, 𝑙}} = {{𝑥, 𝐵}, {𝑥, 𝑙}})
1918sseq1d 3595 . . . . . . . . 9 (𝑘 = 𝐵 → ({{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸))
2019reubidv 3103 . . . . . . . 8 (𝑘 = 𝐵 → (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸))
2116, 20raleqbidv 3129 . . . . . . 7 (𝑘 = 𝐵 → (∀𝑙 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ∀𝑙 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵})∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸))
22 sneq 4135 . . . . . . . . 9 (𝑘 = 𝐶 → {𝑘} = {𝐶})
2322difeq2d 3690 . . . . . . . 8 (𝑘 = 𝐶 → ({𝐴, 𝐵, 𝐶} ∖ {𝑘}) = ({𝐴, 𝐵, 𝐶} ∖ {𝐶}))
24 preq2 4213 . . . . . . . . . . 11 (𝑘 = 𝐶 → {𝑥, 𝑘} = {𝑥, 𝐶})
2524preq1d 4218 . . . . . . . . . 10 (𝑘 = 𝐶 → {{𝑥, 𝑘}, {𝑥, 𝑙}} = {{𝑥, 𝐶}, {𝑥, 𝑙}})
2625sseq1d 3595 . . . . . . . . 9 (𝑘 = 𝐶 → ({{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ {{𝑥, 𝐶}, {𝑥, 𝑙}} ⊆ ran 𝐸))
2726reubidv 3103 . . . . . . . 8 (𝑘 = 𝐶 → (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐶}, {𝑥, 𝑙}} ⊆ ran 𝐸))
2823, 27raleqbidv 3129 . . . . . . 7 (𝑘 = 𝐶 → (∀𝑙 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ∀𝑙 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶})∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐶}, {𝑥, 𝑙}} ⊆ ran 𝐸))
2914, 21, 28raltpg 4183 . . . . . 6 ((𝐴𝑋𝐵𝑌𝐶𝑍) → (∀𝑘 ∈ {𝐴, 𝐵, 𝐶}∀𝑙 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ (∀𝑙 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴})∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸 ∧ ∀𝑙 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵})∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸 ∧ ∀𝑙 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶})∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐶}, {𝑥, 𝑙}} ⊆ ran 𝐸)))
3029adantr 480 . . . . 5 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (∀𝑘 ∈ {𝐴, 𝐵, 𝐶}∀𝑙 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ (∀𝑙 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴})∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸 ∧ ∀𝑙 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵})∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸 ∧ ∀𝑙 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶})∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐶}, {𝑥, 𝑙}} ⊆ ran 𝐸)))
3130adantr 480 . . . 4 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → (∀𝑘 ∈ {𝐴, 𝐵, 𝐶}∀𝑙 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ (∀𝑙 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴})∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸 ∧ ∀𝑙 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵})∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸 ∧ ∀𝑙 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶})∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐶}, {𝑥, 𝑙}} ⊆ ran 𝐸)))
32 tprot 4228 . . . . . . . . . . 11 {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴}
3332a1i 11 . . . . . . . . . 10 ((𝐴𝐵𝐴𝐶𝐵𝐶) → {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴})
3433difeq1d 3689 . . . . . . . . 9 ((𝐴𝐵𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐴}) = ({𝐵, 𝐶, 𝐴} ∖ {𝐴}))
35 necom 2835 . . . . . . . . . . . . 13 (𝐴𝐵𝐵𝐴)
3635biimpi 205 . . . . . . . . . . . 12 (𝐴𝐵𝐵𝐴)
37 necom 2835 . . . . . . . . . . . . 13 (𝐴𝐶𝐶𝐴)
3837biimpi 205 . . . . . . . . . . . 12 (𝐴𝐶𝐶𝐴)
3936, 38anim12i 588 . . . . . . . . . . 11 ((𝐴𝐵𝐴𝐶) → (𝐵𝐴𝐶𝐴))
40393adant3 1074 . . . . . . . . . 10 ((𝐴𝐵𝐴𝐶𝐵𝐶) → (𝐵𝐴𝐶𝐴))
41 diftpsn3 4273 . . . . . . . . . 10 ((𝐵𝐴𝐶𝐴) → ({𝐵, 𝐶, 𝐴} ∖ {𝐴}) = {𝐵, 𝐶})
4240, 41syl 17 . . . . . . . . 9 ((𝐴𝐵𝐴𝐶𝐵𝐶) → ({𝐵, 𝐶, 𝐴} ∖ {𝐴}) = {𝐵, 𝐶})
4334, 42eqtrd 2644 . . . . . . . 8 ((𝐴𝐵𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐴}) = {𝐵, 𝐶})
4443raleqdv 3121 . . . . . . 7 ((𝐴𝐵𝐴𝐶𝐵𝐶) → (∀𝑙 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴})∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ∀𝑙 ∈ {𝐵, 𝐶}∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸))
45 tprot 4228 . . . . . . . . . . . 12 {𝐶, 𝐴, 𝐵} = {𝐴, 𝐵, 𝐶}
4645eqcomi 2619 . . . . . . . . . . 11 {𝐴, 𝐵, 𝐶} = {𝐶, 𝐴, 𝐵}
4746a1i 11 . . . . . . . . . 10 ((𝐴𝐵𝐴𝐶𝐵𝐶) → {𝐴, 𝐵, 𝐶} = {𝐶, 𝐴, 𝐵})
4847difeq1d 3689 . . . . . . . . 9 ((𝐴𝐵𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐵}) = ({𝐶, 𝐴, 𝐵} ∖ {𝐵}))
49 id 22 . . . . . . . . . . . 12 (𝐴𝐵𝐴𝐵)
50 necom 2835 . . . . . . . . . . . . 13 (𝐵𝐶𝐶𝐵)
5150biimpi 205 . . . . . . . . . . . 12 (𝐵𝐶𝐶𝐵)
5249, 51anim12ci 589 . . . . . . . . . . 11 ((𝐴𝐵𝐵𝐶) → (𝐶𝐵𝐴𝐵))
53523adant2 1073 . . . . . . . . . 10 ((𝐴𝐵𝐴𝐶𝐵𝐶) → (𝐶𝐵𝐴𝐵))
54 diftpsn3 4273 . . . . . . . . . 10 ((𝐶𝐵𝐴𝐵) → ({𝐶, 𝐴, 𝐵} ∖ {𝐵}) = {𝐶, 𝐴})
5553, 54syl 17 . . . . . . . . 9 ((𝐴𝐵𝐴𝐶𝐵𝐶) → ({𝐶, 𝐴, 𝐵} ∖ {𝐵}) = {𝐶, 𝐴})
5648, 55eqtrd 2644 . . . . . . . 8 ((𝐴𝐵𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐵}) = {𝐶, 𝐴})
5756raleqdv 3121 . . . . . . 7 ((𝐴𝐵𝐴𝐶𝐵𝐶) → (∀𝑙 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵})∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ∀𝑙 ∈ {𝐶, 𝐴}∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸))
58 diftpsn3 4273 . . . . . . . . 9 ((𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐶}) = {𝐴, 𝐵})
59583adant1 1072 . . . . . . . 8 ((𝐴𝐵𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐶}) = {𝐴, 𝐵})
6059raleqdv 3121 . . . . . . 7 ((𝐴𝐵𝐴𝐶𝐵𝐶) → (∀𝑙 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶})∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐶}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ∀𝑙 ∈ {𝐴, 𝐵}∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐶}, {𝑥, 𝑙}} ⊆ ran 𝐸))
6144, 57, 603anbi123d 1391 . . . . . 6 ((𝐴𝐵𝐴𝐶𝐵𝐶) → ((∀𝑙 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴})∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸 ∧ ∀𝑙 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵})∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸 ∧ ∀𝑙 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶})∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐶}, {𝑥, 𝑙}} ⊆ ran 𝐸) ↔ (∀𝑙 ∈ {𝐵, 𝐶}∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸 ∧ ∀𝑙 ∈ {𝐶, 𝐴}∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸 ∧ ∀𝑙 ∈ {𝐴, 𝐵}∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐶}, {𝑥, 𝑙}} ⊆ ran 𝐸)))
6261adantl 481 . . . . 5 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ((∀𝑙 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴})∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸 ∧ ∀𝑙 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵})∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸 ∧ ∀𝑙 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶})∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐶}, {𝑥, 𝑙}} ⊆ ran 𝐸) ↔ (∀𝑙 ∈ {𝐵, 𝐶}∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸 ∧ ∀𝑙 ∈ {𝐶, 𝐴}∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸 ∧ ∀𝑙 ∈ {𝐴, 𝐵}∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐶}, {𝑥, 𝑙}} ⊆ ran 𝐸)))
6362adantr 480 . . . 4 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → ((∀𝑙 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴})∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸 ∧ ∀𝑙 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵})∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸 ∧ ∀𝑙 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶})∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐶}, {𝑥, 𝑙}} ⊆ ran 𝐸) ↔ (∀𝑙 ∈ {𝐵, 𝐶}∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸 ∧ ∀𝑙 ∈ {𝐶, 𝐴}∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸 ∧ ∀𝑙 ∈ {𝐴, 𝐵}∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐶}, {𝑥, 𝑙}} ⊆ ran 𝐸)))
64 preq2 4213 . . . . . . . . . . . 12 (𝑙 = 𝐵 → {𝑥, 𝑙} = {𝑥, 𝐵})
6564preq2d 4219 . . . . . . . . . . 11 (𝑙 = 𝐵 → {{𝑥, 𝐴}, {𝑥, 𝑙}} = {{𝑥, 𝐴}, {𝑥, 𝐵}})
6665sseq1d 3595 . . . . . . . . . 10 (𝑙 = 𝐵 → ({{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸))
6766reubidv 3103 . . . . . . . . 9 (𝑙 = 𝐵 → (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸))
68 preq2 4213 . . . . . . . . . . . 12 (𝑙 = 𝐶 → {𝑥, 𝑙} = {𝑥, 𝐶})
6968preq2d 4219 . . . . . . . . . . 11 (𝑙 = 𝐶 → {{𝑥, 𝐴}, {𝑥, 𝑙}} = {{𝑥, 𝐴}, {𝑥, 𝐶}})
7069sseq1d 3595 . . . . . . . . . 10 (𝑙 = 𝐶 → ({{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ {{𝑥, 𝐴}, {𝑥, 𝐶}} ⊆ ran 𝐸))
7170reubidv 3103 . . . . . . . . 9 (𝑙 = 𝐶 → (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝐶}} ⊆ ran 𝐸))
7267, 71ralprg 4181 . . . . . . . 8 ((𝐵𝑌𝐶𝑍) → (∀𝑙 ∈ {𝐵, 𝐶}∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 ∧ ∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝐶}} ⊆ ran 𝐸)))
73723adant1 1072 . . . . . . 7 ((𝐴𝑋𝐵𝑌𝐶𝑍) → (∀𝑙 ∈ {𝐵, 𝐶}∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 ∧ ∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝐶}} ⊆ ran 𝐸)))
7468preq2d 4219 . . . . . . . . . . . 12 (𝑙 = 𝐶 → {{𝑥, 𝐵}, {𝑥, 𝑙}} = {{𝑥, 𝐵}, {𝑥, 𝐶}})
7574sseq1d 3595 . . . . . . . . . . 11 (𝑙 = 𝐶 → ({{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ {{𝑥, 𝐵}, {𝑥, 𝐶}} ⊆ ran 𝐸))
7675reubidv 3103 . . . . . . . . . 10 (𝑙 = 𝐶 → (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐵}, {𝑥, 𝐶}} ⊆ ran 𝐸))
77 preq2 4213 . . . . . . . . . . . . 13 (𝑙 = 𝐴 → {𝑥, 𝑙} = {𝑥, 𝐴})
7877preq2d 4219 . . . . . . . . . . . 12 (𝑙 = 𝐴 → {{𝑥, 𝐵}, {𝑥, 𝑙}} = {{𝑥, 𝐵}, {𝑥, 𝐴}})
7978sseq1d 3595 . . . . . . . . . . 11 (𝑙 = 𝐴 → ({{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ {{𝑥, 𝐵}, {𝑥, 𝐴}} ⊆ ran 𝐸))
8079reubidv 3103 . . . . . . . . . 10 (𝑙 = 𝐴 → (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐵}, {𝑥, 𝐴}} ⊆ ran 𝐸))
8176, 80ralprg 4181 . . . . . . . . 9 ((𝐶𝑍𝐴𝑋) → (∀𝑙 ∈ {𝐶, 𝐴}∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐵}, {𝑥, 𝐶}} ⊆ ran 𝐸 ∧ ∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐵}, {𝑥, 𝐴}} ⊆ ran 𝐸)))
8281ancoms 468 . . . . . . . 8 ((𝐴𝑋𝐶𝑍) → (∀𝑙 ∈ {𝐶, 𝐴}∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐵}, {𝑥, 𝐶}} ⊆ ran 𝐸 ∧ ∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐵}, {𝑥, 𝐴}} ⊆ ran 𝐸)))
83823adant2 1073 . . . . . . 7 ((𝐴𝑋𝐵𝑌𝐶𝑍) → (∀𝑙 ∈ {𝐶, 𝐴}∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐵}, {𝑥, 𝐶}} ⊆ ran 𝐸 ∧ ∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐵}, {𝑥, 𝐴}} ⊆ ran 𝐸)))
8477preq2d 4219 . . . . . . . . . . 11 (𝑙 = 𝐴 → {{𝑥, 𝐶}, {𝑥, 𝑙}} = {{𝑥, 𝐶}, {𝑥, 𝐴}})
8584sseq1d 3595 . . . . . . . . . 10 (𝑙 = 𝐴 → ({{𝑥, 𝐶}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ {{𝑥, 𝐶}, {𝑥, 𝐴}} ⊆ ran 𝐸))
8685reubidv 3103 . . . . . . . . 9 (𝑙 = 𝐴 → (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐶}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐶}, {𝑥, 𝐴}} ⊆ ran 𝐸))
8764preq2d 4219 . . . . . . . . . . 11 (𝑙 = 𝐵 → {{𝑥, 𝐶}, {𝑥, 𝑙}} = {{𝑥, 𝐶}, {𝑥, 𝐵}})
8887sseq1d 3595 . . . . . . . . . 10 (𝑙 = 𝐵 → ({{𝑥, 𝐶}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ {{𝑥, 𝐶}, {𝑥, 𝐵}} ⊆ ran 𝐸))
8988reubidv 3103 . . . . . . . . 9 (𝑙 = 𝐵 → (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐶}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐶}, {𝑥, 𝐵}} ⊆ ran 𝐸))
9086, 89ralprg 4181 . . . . . . . 8 ((𝐴𝑋𝐵𝑌) → (∀𝑙 ∈ {𝐴, 𝐵}∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐶}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐶}, {𝑥, 𝐴}} ⊆ ran 𝐸 ∧ ∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐶}, {𝑥, 𝐵}} ⊆ ran 𝐸)))
91903adant3 1074 . . . . . . 7 ((𝐴𝑋𝐵𝑌𝐶𝑍) → (∀𝑙 ∈ {𝐴, 𝐵}∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐶}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐶}, {𝑥, 𝐴}} ⊆ ran 𝐸 ∧ ∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐶}, {𝑥, 𝐵}} ⊆ ran 𝐸)))
9273, 83, 913anbi123d 1391 . . . . . 6 ((𝐴𝑋𝐵𝑌𝐶𝑍) → ((∀𝑙 ∈ {𝐵, 𝐶}∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸 ∧ ∀𝑙 ∈ {𝐶, 𝐴}∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸 ∧ ∀𝑙 ∈ {𝐴, 𝐵}∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐶}, {𝑥, 𝑙}} ⊆ ran 𝐸) ↔ ((∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 ∧ ∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝐶}} ⊆ ran 𝐸) ∧ (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐵}, {𝑥, 𝐶}} ⊆ ran 𝐸 ∧ ∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐵}, {𝑥, 𝐴}} ⊆ ran 𝐸) ∧ (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐶}, {𝑥, 𝐴}} ⊆ ran 𝐸 ∧ ∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐶}, {𝑥, 𝐵}} ⊆ ran 𝐸))))
9392adantr 480 . . . . 5 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ((∀𝑙 ∈ {𝐵, 𝐶}∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸 ∧ ∀𝑙 ∈ {𝐶, 𝐴}∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸 ∧ ∀𝑙 ∈ {𝐴, 𝐵}∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐶}, {𝑥, 𝑙}} ⊆ ran 𝐸) ↔ ((∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 ∧ ∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝐶}} ⊆ ran 𝐸) ∧ (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐵}, {𝑥, 𝐶}} ⊆ ran 𝐸 ∧ ∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐵}, {𝑥, 𝐴}} ⊆ ran 𝐸) ∧ (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐶}, {𝑥, 𝐴}} ⊆ ran 𝐸 ∧ ∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐶}, {𝑥, 𝐵}} ⊆ ran 𝐸))))
9493adantr 480 . . . 4 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → ((∀𝑙 ∈ {𝐵, 𝐶}∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸 ∧ ∀𝑙 ∈ {𝐶, 𝐴}∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸 ∧ ∀𝑙 ∈ {𝐴, 𝐵}∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐶}, {𝑥, 𝑙}} ⊆ ran 𝐸) ↔ ((∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 ∧ ∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝐶}} ⊆ ran 𝐸) ∧ (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐵}, {𝑥, 𝐶}} ⊆ ran 𝐸 ∧ ∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐵}, {𝑥, 𝐴}} ⊆ ran 𝐸) ∧ (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐶}, {𝑥, 𝐴}} ⊆ ran 𝐸 ∧ ∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐶}, {𝑥, 𝐵}} ⊆ ran 𝐸))))
9531, 63, 943bitrd 293 . . 3 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → (∀𝑘 ∈ {𝐴, 𝐵, 𝐶}∀𝑙 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ((∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 ∧ ∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝐶}} ⊆ ran 𝐸) ∧ (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐵}, {𝑥, 𝐶}} ⊆ ran 𝐸 ∧ ∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐵}, {𝑥, 𝐴}} ⊆ ran 𝐸) ∧ (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐶}, {𝑥, 𝐴}} ⊆ ran 𝐸 ∧ ∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐶}, {𝑥, 𝐵}} ⊆ ran 𝐸))))
96 frgra3vlem2 26528 . . . . . . 7 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ({𝐴, 𝐵, 𝐶} USGrph 𝐸 → (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 ↔ ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸))))
9796imp 444 . . . . . 6 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 ↔ ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸)))
98 3ancomb 1040 . . . . . . . 8 ((𝐴𝑋𝐵𝑌𝐶𝑍) ↔ (𝐴𝑋𝐶𝑍𝐵𝑌))
99 3ancoma 1038 . . . . . . . . 9 ((𝐴𝐵𝐴𝐶𝐵𝐶) ↔ (𝐴𝐶𝐴𝐵𝐵𝐶))
100 biid 250 . . . . . . . . . 10 (𝐴𝐶𝐴𝐶)
101 biid 250 . . . . . . . . . 10 (𝐴𝐵𝐴𝐵)
102100, 101, 503anbi123i 1244 . . . . . . . . 9 ((𝐴𝐶𝐴𝐵𝐵𝐶) ↔ (𝐴𝐶𝐴𝐵𝐶𝐵))
10399, 102bitri 263 . . . . . . . 8 ((𝐴𝐵𝐴𝐶𝐵𝐶) ↔ (𝐴𝐶𝐴𝐵𝐶𝐵))
10498, 103anbi12i 729 . . . . . . 7 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ↔ ((𝐴𝑋𝐶𝑍𝐵𝑌) ∧ (𝐴𝐶𝐴𝐵𝐶𝐵)))
105 tpcomb 4230 . . . . . . . 8 {𝐴, 𝐵, 𝐶} = {𝐴, 𝐶, 𝐵}
106105breq1i 4590 . . . . . . 7 ({𝐴, 𝐵, 𝐶} USGrph 𝐸 ↔ {𝐴, 𝐶, 𝐵} USGrph 𝐸)
107 reueq1 3117 . . . . . . . . 9 ({𝐴, 𝐵, 𝐶} = {𝐴, 𝐶, 𝐵} → (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝐶}} ⊆ ran 𝐸 ↔ ∃!𝑥 ∈ {𝐴, 𝐶, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐶}} ⊆ ran 𝐸))
108105, 107mp1i 13 . . . . . . . 8 ((((𝐴𝑋𝐶𝑍𝐵𝑌) ∧ (𝐴𝐶𝐴𝐵𝐶𝐵)) ∧ {𝐴, 𝐶, 𝐵} USGrph 𝐸) → (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝐶}} ⊆ ran 𝐸 ↔ ∃!𝑥 ∈ {𝐴, 𝐶, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐶}} ⊆ ran 𝐸))
109 frgra3vlem2 26528 . . . . . . . . 9 (((𝐴𝑋𝐶𝑍𝐵𝑌) ∧ (𝐴𝐶𝐴𝐵𝐶𝐵)) → ({𝐴, 𝐶, 𝐵} USGrph 𝐸 → (∃!𝑥 ∈ {𝐴, 𝐶, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐶}} ⊆ ran 𝐸 ↔ ({𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸))))
110109imp 444 . . . . . . . 8 ((((𝐴𝑋𝐶𝑍𝐵𝑌) ∧ (𝐴𝐶𝐴𝐵𝐶𝐵)) ∧ {𝐴, 𝐶, 𝐵} USGrph 𝐸) → (∃!𝑥 ∈ {𝐴, 𝐶, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐶}} ⊆ ran 𝐸 ↔ ({𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)))
111108, 110bitrd 267 . . . . . . 7 ((((𝐴𝑋𝐶𝑍𝐵𝑌) ∧ (𝐴𝐶𝐴𝐵𝐶𝐵)) ∧ {𝐴, 𝐶, 𝐵} USGrph 𝐸) → (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝐶}} ⊆ ran 𝐸 ↔ ({𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)))
112104, 106, 111syl2anb 495 . . . . . 6 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝐶}} ⊆ ran 𝐸 ↔ ({𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)))
11397, 112anbi12d 743 . . . . 5 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → ((∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 ∧ ∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝐶}} ⊆ ran 𝐸) ↔ (({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸) ∧ ({𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸))))
114 3anrot 1036 . . . . . . . 8 ((𝐴𝑋𝐵𝑌𝐶𝑍) ↔ (𝐵𝑌𝐶𝑍𝐴𝑋))
115 3anrot 1036 . . . . . . . . 9 ((𝐵𝐶𝐴𝐵𝐴𝐶) ↔ (𝐴𝐵𝐴𝐶𝐵𝐶))
116 biid 250 . . . . . . . . . 10 (𝐵𝐶𝐵𝐶)
117116, 35, 373anbi123i 1244 . . . . . . . . 9 ((𝐵𝐶𝐴𝐵𝐴𝐶) ↔ (𝐵𝐶𝐵𝐴𝐶𝐴))
118115, 117bitr3i 265 . . . . . . . 8 ((𝐴𝐵𝐴𝐶𝐵𝐶) ↔ (𝐵𝐶𝐵𝐴𝐶𝐴))
119114, 118anbi12i 729 . . . . . . 7 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ↔ ((𝐵𝑌𝐶𝑍𝐴𝑋) ∧ (𝐵𝐶𝐵𝐴𝐶𝐴)))
12032breq1i 4590 . . . . . . 7 ({𝐴, 𝐵, 𝐶} USGrph 𝐸 ↔ {𝐵, 𝐶, 𝐴} USGrph 𝐸)
121 reueq1 3117 . . . . . . . . 9 ({𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴} → (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐵}, {𝑥, 𝐶}} ⊆ ran 𝐸 ↔ ∃!𝑥 ∈ {𝐵, 𝐶, 𝐴} {{𝑥, 𝐵}, {𝑥, 𝐶}} ⊆ ran 𝐸))
12232, 121mp1i 13 . . . . . . . 8 ((((𝐵𝑌𝐶𝑍𝐴𝑋) ∧ (𝐵𝐶𝐵𝐴𝐶𝐴)) ∧ {𝐵, 𝐶, 𝐴} USGrph 𝐸) → (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐵}, {𝑥, 𝐶}} ⊆ ran 𝐸 ↔ ∃!𝑥 ∈ {𝐵, 𝐶, 𝐴} {{𝑥, 𝐵}, {𝑥, 𝐶}} ⊆ ran 𝐸))
123 frgra3vlem2 26528 . . . . . . . . 9 (((𝐵𝑌𝐶𝑍𝐴𝑋) ∧ (𝐵𝐶𝐵𝐴𝐶𝐴)) → ({𝐵, 𝐶, 𝐴} USGrph 𝐸 → (∃!𝑥 ∈ {𝐵, 𝐶, 𝐴} {{𝑥, 𝐵}, {𝑥, 𝐶}} ⊆ ran 𝐸 ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸))))
124123imp 444 . . . . . . . 8 ((((𝐵𝑌𝐶𝑍𝐴𝑋) ∧ (𝐵𝐶𝐵𝐴𝐶𝐴)) ∧ {𝐵, 𝐶, 𝐴} USGrph 𝐸) → (∃!𝑥 ∈ {𝐵, 𝐶, 𝐴} {{𝑥, 𝐵}, {𝑥, 𝐶}} ⊆ ran 𝐸 ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸)))
125122, 124bitrd 267 . . . . . . 7 ((((𝐵𝑌𝐶𝑍𝐴𝑋) ∧ (𝐵𝐶𝐵𝐴𝐶𝐴)) ∧ {𝐵, 𝐶, 𝐴} USGrph 𝐸) → (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐵}, {𝑥, 𝐶}} ⊆ ran 𝐸 ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸)))
126119, 120, 125syl2anb 495 . . . . . 6 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐵}, {𝑥, 𝐶}} ⊆ ran 𝐸 ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸)))
127 3ancoma 1038 . . . . . . . 8 ((𝐴𝑋𝐵𝑌𝐶𝑍) ↔ (𝐵𝑌𝐴𝑋𝐶𝑍))
128 3ancomb 1040 . . . . . . . . 9 ((𝐴𝐵𝐴𝐶𝐵𝐶) ↔ (𝐴𝐵𝐵𝐶𝐴𝐶))
12935, 116, 1003anbi123i 1244 . . . . . . . . 9 ((𝐴𝐵𝐵𝐶𝐴𝐶) ↔ (𝐵𝐴𝐵𝐶𝐴𝐶))
130128, 129bitri 263 . . . . . . . 8 ((𝐴𝐵𝐴𝐶𝐵𝐶) ↔ (𝐵𝐴𝐵𝐶𝐴𝐶))
131127, 130anbi12i 729 . . . . . . 7 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ↔ ((𝐵𝑌𝐴𝑋𝐶𝑍) ∧ (𝐵𝐴𝐵𝐶𝐴𝐶)))
132 tpcoma 4229 . . . . . . . 8 {𝐴, 𝐵, 𝐶} = {𝐵, 𝐴, 𝐶}
133132breq1i 4590 . . . . . . 7 ({𝐴, 𝐵, 𝐶} USGrph 𝐸 ↔ {𝐵, 𝐴, 𝐶} USGrph 𝐸)
134 reueq1 3117 . . . . . . . . 9 ({𝐴, 𝐵, 𝐶} = {𝐵, 𝐴, 𝐶} → (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐵}, {𝑥, 𝐴}} ⊆ ran 𝐸 ↔ ∃!𝑥 ∈ {𝐵, 𝐴, 𝐶} {{𝑥, 𝐵}, {𝑥, 𝐴}} ⊆ ran 𝐸))
135132, 134mp1i 13 . . . . . . . 8 ((((𝐵𝑌𝐴𝑋𝐶𝑍) ∧ (𝐵𝐴𝐵𝐶𝐴𝐶)) ∧ {𝐵, 𝐴, 𝐶} USGrph 𝐸) → (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐵}, {𝑥, 𝐴}} ⊆ ran 𝐸 ↔ ∃!𝑥 ∈ {𝐵, 𝐴, 𝐶} {{𝑥, 𝐵}, {𝑥, 𝐴}} ⊆ ran 𝐸))
136 frgra3vlem2 26528 . . . . . . . . 9 (((𝐵𝑌𝐴𝑋𝐶𝑍) ∧ (𝐵𝐴𝐵𝐶𝐴𝐶)) → ({𝐵, 𝐴, 𝐶} USGrph 𝐸 → (∃!𝑥 ∈ {𝐵, 𝐴, 𝐶} {{𝑥, 𝐵}, {𝑥, 𝐴}} ⊆ ran 𝐸 ↔ ({𝐶, 𝐵} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))))
137136imp 444 . . . . . . . 8 ((((𝐵𝑌𝐴𝑋𝐶𝑍) ∧ (𝐵𝐴𝐵𝐶𝐴𝐶)) ∧ {𝐵, 𝐴, 𝐶} USGrph 𝐸) → (∃!𝑥 ∈ {𝐵, 𝐴, 𝐶} {{𝑥, 𝐵}, {𝑥, 𝐴}} ⊆ ran 𝐸 ↔ ({𝐶, 𝐵} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)))
138135, 137bitrd 267 . . . . . . 7 ((((𝐵𝑌𝐴𝑋𝐶𝑍) ∧ (𝐵𝐴𝐵𝐶𝐴𝐶)) ∧ {𝐵, 𝐴, 𝐶} USGrph 𝐸) → (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐵}, {𝑥, 𝐴}} ⊆ ran 𝐸 ↔ ({𝐶, 𝐵} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)))
139131, 133, 138syl2anb 495 . . . . . 6 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐵}, {𝑥, 𝐴}} ⊆ ran 𝐸 ↔ ({𝐶, 𝐵} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)))
140126, 139anbi12d 743 . . . . 5 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → ((∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐵}, {𝑥, 𝐶}} ⊆ ran 𝐸 ∧ ∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐵}, {𝑥, 𝐴}} ⊆ ran 𝐸) ↔ (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸) ∧ ({𝐶, 𝐵} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))))
141 3anrot 1036 . . . . . . . . 9 ((𝐶𝑍𝐴𝑋𝐵𝑌) ↔ (𝐴𝑋𝐵𝑌𝐶𝑍))
142141biimpri 217 . . . . . . . 8 ((𝐴𝑋𝐵𝑌𝐶𝑍) → (𝐶𝑍𝐴𝑋𝐵𝑌))
143 3anrot 1036 . . . . . . . . . 10 ((𝐴𝐵𝐴𝐶𝐵𝐶) ↔ (𝐴𝐶𝐵𝐶𝐴𝐵))
14437, 50, 1013anbi123i 1244 . . . . . . . . . 10 ((𝐴𝐶𝐵𝐶𝐴𝐵) ↔ (𝐶𝐴𝐶𝐵𝐴𝐵))
145143, 144bitri 263 . . . . . . . . 9 ((𝐴𝐵𝐴𝐶𝐵𝐶) ↔ (𝐶𝐴𝐶𝐵𝐴𝐵))
146145biimpi 205 . . . . . . . 8 ((𝐴𝐵𝐴𝐶𝐵𝐶) → (𝐶𝐴𝐶𝐵𝐴𝐵))
147142, 146anim12i 588 . . . . . . 7 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ((𝐶𝑍𝐴𝑋𝐵𝑌) ∧ (𝐶𝐴𝐶𝐵𝐴𝐵)))
14846breq1i 4590 . . . . . . . 8 ({𝐴, 𝐵, 𝐶} USGrph 𝐸 ↔ {𝐶, 𝐴, 𝐵} USGrph 𝐸)
149148biimpi 205 . . . . . . 7 ({𝐴, 𝐵, 𝐶} USGrph 𝐸 → {𝐶, 𝐴, 𝐵} USGrph 𝐸)
150 reueq1 3117 . . . . . . . . 9 ({𝐴, 𝐵, 𝐶} = {𝐶, 𝐴, 𝐵} → (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐶}, {𝑥, 𝐴}} ⊆ ran 𝐸 ↔ ∃!𝑥 ∈ {𝐶, 𝐴, 𝐵} {{𝑥, 𝐶}, {𝑥, 𝐴}} ⊆ ran 𝐸))
15146, 150mp1i 13 . . . . . . . 8 ((((𝐶𝑍𝐴𝑋𝐵𝑌) ∧ (𝐶𝐴𝐶𝐵𝐴𝐵)) ∧ {𝐶, 𝐴, 𝐵} USGrph 𝐸) → (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐶}, {𝑥, 𝐴}} ⊆ ran 𝐸 ↔ ∃!𝑥 ∈ {𝐶, 𝐴, 𝐵} {{𝑥, 𝐶}, {𝑥, 𝐴}} ⊆ ran 𝐸))
152 frgra3vlem2 26528 . . . . . . . . 9 (((𝐶𝑍𝐴𝑋𝐵𝑌) ∧ (𝐶𝐴𝐶𝐵𝐴𝐵)) → ({𝐶, 𝐴, 𝐵} USGrph 𝐸 → (∃!𝑥 ∈ {𝐶, 𝐴, 𝐵} {{𝑥, 𝐶}, {𝑥, 𝐴}} ⊆ ran 𝐸 ↔ ({𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐵, 𝐴} ∈ ran 𝐸))))
153152imp 444 . . . . . . . 8 ((((𝐶𝑍𝐴𝑋𝐵𝑌) ∧ (𝐶𝐴𝐶𝐵𝐴𝐵)) ∧ {𝐶, 𝐴, 𝐵} USGrph 𝐸) → (∃!𝑥 ∈ {𝐶, 𝐴, 𝐵} {{𝑥, 𝐶}, {𝑥, 𝐴}} ⊆ ran 𝐸 ↔ ({𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐵, 𝐴} ∈ ran 𝐸)))
154151, 153bitrd 267 . . . . . . 7 ((((𝐶𝑍𝐴𝑋𝐵𝑌) ∧ (𝐶𝐴𝐶𝐵𝐴𝐵)) ∧ {𝐶, 𝐴, 𝐵} USGrph 𝐸) → (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐶}, {𝑥, 𝐴}} ⊆ ran 𝐸 ↔ ({𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐵, 𝐴} ∈ ran 𝐸)))
155147, 149, 154syl2an 493 . . . . . 6 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐶}, {𝑥, 𝐴}} ⊆ ran 𝐸 ↔ ({𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐵, 𝐴} ∈ ran 𝐸)))
156 3anrev 1042 . . . . . . . . 9 ((𝐴𝑋𝐵𝑌𝐶𝑍) ↔ (𝐶𝑍𝐵𝑌𝐴𝑋))
157156biimpi 205 . . . . . . . 8 ((𝐴𝑋𝐵𝑌𝐶𝑍) → (𝐶𝑍𝐵𝑌𝐴𝑋))
15850, 37, 353anbi123i 1244 . . . . . . . . . 10 ((𝐵𝐶𝐴𝐶𝐴𝐵) ↔ (𝐶𝐵𝐶𝐴𝐵𝐴))
159158biimpi 205 . . . . . . . . 9 ((𝐵𝐶𝐴𝐶𝐴𝐵) → (𝐶𝐵𝐶𝐴𝐵𝐴))
1601593com13 1262 . . . . . . . 8 ((𝐴𝐵𝐴𝐶𝐵𝐶) → (𝐶𝐵𝐶𝐴𝐵𝐴))
161157, 160anim12i 588 . . . . . . 7 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ((𝐶𝑍𝐵𝑌𝐴𝑋) ∧ (𝐶𝐵𝐶𝐴𝐵𝐴)))
162 tpcoma 4229 . . . . . . . . . 10 {𝐵, 𝐶, 𝐴} = {𝐶, 𝐵, 𝐴}
16332, 162eqtri 2632 . . . . . . . . 9 {𝐴, 𝐵, 𝐶} = {𝐶, 𝐵, 𝐴}
164163breq1i 4590 . . . . . . . 8 ({𝐴, 𝐵, 𝐶} USGrph 𝐸 ↔ {𝐶, 𝐵, 𝐴} USGrph 𝐸)
165164biimpi 205 . . . . . . 7 ({𝐴, 𝐵, 𝐶} USGrph 𝐸 → {𝐶, 𝐵, 𝐴} USGrph 𝐸)
166 reueq1 3117 . . . . . . . . 9 ({𝐴, 𝐵, 𝐶} = {𝐶, 𝐵, 𝐴} → (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐶}, {𝑥, 𝐵}} ⊆ ran 𝐸 ↔ ∃!𝑥 ∈ {𝐶, 𝐵, 𝐴} {{𝑥, 𝐶}, {𝑥, 𝐵}} ⊆ ran 𝐸))
167163, 166mp1i 13 . . . . . . . 8 ((((𝐶𝑍𝐵𝑌𝐴𝑋) ∧ (𝐶𝐵𝐶𝐴𝐵𝐴)) ∧ {𝐶, 𝐵, 𝐴} USGrph 𝐸) → (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐶}, {𝑥, 𝐵}} ⊆ ran 𝐸 ↔ ∃!𝑥 ∈ {𝐶, 𝐵, 𝐴} {{𝑥, 𝐶}, {𝑥, 𝐵}} ⊆ ran 𝐸))
168 frgra3vlem2 26528 . . . . . . . . 9 (((𝐶𝑍𝐵𝑌𝐴𝑋) ∧ (𝐶𝐵𝐶𝐴𝐵𝐴)) → ({𝐶, 𝐵, 𝐴} USGrph 𝐸 → (∃!𝑥 ∈ {𝐶, 𝐵, 𝐴} {{𝑥, 𝐶}, {𝑥, 𝐵}} ⊆ ran 𝐸 ↔ ({𝐴, 𝐶} ∈ ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸))))
169168imp 444 . . . . . . . 8 ((((𝐶𝑍𝐵𝑌𝐴𝑋) ∧ (𝐶𝐵𝐶𝐴𝐵𝐴)) ∧ {𝐶, 𝐵, 𝐴} USGrph 𝐸) → (∃!𝑥 ∈ {𝐶, 𝐵, 𝐴} {{𝑥, 𝐶}, {𝑥, 𝐵}} ⊆ ran 𝐸 ↔ ({𝐴, 𝐶} ∈ ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸)))
170167, 169bitrd 267 . . . . . . 7 ((((𝐶𝑍𝐵𝑌𝐴𝑋) ∧ (𝐶𝐵𝐶𝐴𝐵𝐴)) ∧ {𝐶, 𝐵, 𝐴} USGrph 𝐸) → (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐶}, {𝑥, 𝐵}} ⊆ ran 𝐸 ↔ ({𝐴, 𝐶} ∈ ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸)))
171161, 165, 170syl2an 493 . . . . . 6 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐶}, {𝑥, 𝐵}} ⊆ ran 𝐸 ↔ ({𝐴, 𝐶} ∈ ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸)))
172155, 171anbi12d 743 . . . . 5 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → ((∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐶}, {𝑥, 𝐴}} ⊆ ran 𝐸 ∧ ∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐶}, {𝑥, 𝐵}} ⊆ ran 𝐸) ↔ (({𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐵, 𝐴} ∈ ran 𝐸) ∧ ({𝐴, 𝐶} ∈ ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸))))
173113, 140, 1723anbi123d 1391 . . . 4 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → (((∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 ∧ ∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝐶}} ⊆ ran 𝐸) ∧ (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐵}, {𝑥, 𝐶}} ⊆ ran 𝐸 ∧ ∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐵}, {𝑥, 𝐴}} ⊆ ran 𝐸) ∧ (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐶}, {𝑥, 𝐴}} ⊆ ran 𝐸 ∧ ∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐶}, {𝑥, 𝐵}} ⊆ ran 𝐸)) ↔ ((({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸) ∧ ({𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) ∧ (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸) ∧ ({𝐶, 𝐵} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) ∧ (({𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐵, 𝐴} ∈ ran 𝐸) ∧ ({𝐴, 𝐶} ∈ ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸)))))
174 prcom 4211 . . . . . . . . . . 11 {𝐵, 𝐶} = {𝐶, 𝐵}
175174eleq1i 2679 . . . . . . . . . 10 ({𝐵, 𝐶} ∈ ran 𝐸 ↔ {𝐶, 𝐵} ∈ ran 𝐸)
176175anbi2i 726 . . . . . . . . 9 (({𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) ↔ ({𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸))
177176anbi2i 726 . . . . . . . 8 ((({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸) ∧ ({𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) ↔ (({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸) ∧ ({𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸)))
178 anandir 868 . . . . . . . 8 ((({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐵, 𝐴} ∈ ran 𝐸) ∧ {𝐶, 𝐵} ∈ ran 𝐸) ↔ (({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸) ∧ ({𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸)))
179177, 178bitr4i 266 . . . . . . 7 ((({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸) ∧ ({𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) ↔ (({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐵, 𝐴} ∈ ran 𝐸) ∧ {𝐶, 𝐵} ∈ ran 𝐸))
180 prcom 4211 . . . . . . . . . . 11 {𝐶, 𝐴} = {𝐴, 𝐶}
181180eleq1i 2679 . . . . . . . . . 10 ({𝐶, 𝐴} ∈ ran 𝐸 ↔ {𝐴, 𝐶} ∈ ran 𝐸)
182181anbi2i 726 . . . . . . . . 9 (({𝐶, 𝐵} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) ↔ ({𝐶, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸))
183182anbi2i 726 . . . . . . . 8 ((({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸) ∧ ({𝐶, 𝐵} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) ↔ (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸) ∧ ({𝐶, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸)))
184 anandir 868 . . . . . . . 8 ((({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸) ∧ {𝐴, 𝐶} ∈ ran 𝐸) ↔ (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸) ∧ ({𝐶, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸)))
185183, 184bitr4i 266 . . . . . . 7 ((({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸) ∧ ({𝐶, 𝐵} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) ↔ (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸) ∧ {𝐴, 𝐶} ∈ ran 𝐸))
186 prcom 4211 . . . . . . . . . . 11 {𝐴, 𝐵} = {𝐵, 𝐴}
187186eleq1i 2679 . . . . . . . . . 10 ({𝐴, 𝐵} ∈ ran 𝐸 ↔ {𝐵, 𝐴} ∈ ran 𝐸)
188187anbi2i 726 . . . . . . . . 9 (({𝐴, 𝐶} ∈ ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸) ↔ ({𝐴, 𝐶} ∈ ran 𝐸 ∧ {𝐵, 𝐴} ∈ ran 𝐸))
189188anbi2i 726 . . . . . . . 8 ((({𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐵, 𝐴} ∈ ran 𝐸) ∧ ({𝐴, 𝐶} ∈ ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸)) ↔ (({𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐵, 𝐴} ∈ ran 𝐸) ∧ ({𝐴, 𝐶} ∈ ran 𝐸 ∧ {𝐵, 𝐴} ∈ ran 𝐸)))
190 anandir 868 . . . . . . . 8 ((({𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸) ∧ {𝐵, 𝐴} ∈ ran 𝐸) ↔ (({𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐵, 𝐴} ∈ ran 𝐸) ∧ ({𝐴, 𝐶} ∈ ran 𝐸 ∧ {𝐵, 𝐴} ∈ ran 𝐸)))
191189, 190bitr4i 266 . . . . . . 7 ((({𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐵, 𝐴} ∈ ran 𝐸) ∧ ({𝐴, 𝐶} ∈ ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸)) ↔ (({𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸) ∧ {𝐵, 𝐴} ∈ ran 𝐸))
192179, 185, 1913anbi123i 1244 . . . . . 6 (((({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸) ∧ ({𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) ∧ (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸) ∧ ({𝐶, 𝐵} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) ∧ (({𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐵, 𝐴} ∈ ran 𝐸) ∧ ({𝐴, 𝐶} ∈ ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸))) ↔ ((({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐵, 𝐴} ∈ ran 𝐸) ∧ {𝐶, 𝐵} ∈ ran 𝐸) ∧ (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸) ∧ {𝐴, 𝐶} ∈ ran 𝐸) ∧ (({𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸) ∧ {𝐵, 𝐴} ∈ ran 𝐸)))
193 df-3an 1033 . . . . . . . 8 (({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸) ↔ (({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐵, 𝐴} ∈ ran 𝐸) ∧ {𝐶, 𝐵} ∈ ran 𝐸))
194 3anrot 1036 . . . . . . . . 9 (({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸) ↔ ({𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))
195 prcom 4211 . . . . . . . . . . 11 {𝐵, 𝐴} = {𝐴, 𝐵}
196195eleq1i 2679 . . . . . . . . . 10 ({𝐵, 𝐴} ∈ ran 𝐸 ↔ {𝐴, 𝐵} ∈ ran 𝐸)
197 prcom 4211 . . . . . . . . . . 11 {𝐶, 𝐵} = {𝐵, 𝐶}
198197eleq1i 2679 . . . . . . . . . 10 ({𝐶, 𝐵} ∈ ran 𝐸 ↔ {𝐵, 𝐶} ∈ ran 𝐸)
199 biid 250 . . . . . . . . . 10 ({𝐶, 𝐴} ∈ ran 𝐸 ↔ {𝐶, 𝐴} ∈ ran 𝐸)
200196, 198, 1993anbi123i 1244 . . . . . . . . 9 (({𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))
201194, 200bitri 263 . . . . . . . 8 (({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸) ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))
202193, 201bitr3i 265 . . . . . . 7 ((({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐵, 𝐴} ∈ ran 𝐸) ∧ {𝐶, 𝐵} ∈ ran 𝐸) ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))
203 df-3an 1033 . . . . . . . 8 (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸) ↔ (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸) ∧ {𝐴, 𝐶} ∈ ran 𝐸))
204 biid 250 . . . . . . . . 9 ({𝐴, 𝐵} ∈ ran 𝐸 ↔ {𝐴, 𝐵} ∈ ran 𝐸)
205 prcom 4211 . . . . . . . . . 10 {𝐴, 𝐶} = {𝐶, 𝐴}
206205eleq1i 2679 . . . . . . . . 9 ({𝐴, 𝐶} ∈ ran 𝐸 ↔ {𝐶, 𝐴} ∈ ran 𝐸)
207204, 198, 2063anbi123i 1244 . . . . . . . 8 (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸) ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))
208203, 207bitr3i 265 . . . . . . 7 ((({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸) ∧ {𝐴, 𝐶} ∈ ran 𝐸) ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))
209 df-3an 1033 . . . . . . . 8 (({𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸 ∧ {𝐵, 𝐴} ∈ ran 𝐸) ↔ (({𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸) ∧ {𝐵, 𝐴} ∈ ran 𝐸))
210 3anrot 1036 . . . . . . . . 9 (({𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸 ∧ {𝐵, 𝐴} ∈ ran 𝐸) ↔ ({𝐴, 𝐶} ∈ ran 𝐸 ∧ {𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸))
211 3anrot 1036 . . . . . . . . 9 (({𝐴, 𝐶} ∈ ran 𝐸 ∧ {𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) ↔ ({𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸))
212 biid 250 . . . . . . . . . 10 ({𝐵, 𝐶} ∈ ran 𝐸 ↔ {𝐵, 𝐶} ∈ ran 𝐸)
213196, 212, 2063anbi123i 1244 . . . . . . . . 9 (({𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸) ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))
214210, 211, 2133bitri 285 . . . . . . . 8 (({𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸 ∧ {𝐵, 𝐴} ∈ ran 𝐸) ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))
215209, 214bitr3i 265 . . . . . . 7 ((({𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸) ∧ {𝐵, 𝐴} ∈ ran 𝐸) ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))
216202, 208, 2153anbi123i 1244 . . . . . 6 (((({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐵, 𝐴} ∈ ran 𝐸) ∧ {𝐶, 𝐵} ∈ ran 𝐸) ∧ (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸) ∧ {𝐴, 𝐶} ∈ ran 𝐸) ∧ (({𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸) ∧ {𝐵, 𝐴} ∈ ran 𝐸)) ↔ (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)))
217 df-3an 1033 . . . . . . 7 ((({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) ↔ ((({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)))
218 anabs1 846 . . . . . . 7 (((({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) ↔ (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)))
219 anidm 674 . . . . . . 7 ((({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))
220217, 218, 2193bitri 285 . . . . . 6 ((({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))
221192, 216, 2203bitri 285 . . . . 5 (((({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸) ∧ ({𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) ∧ (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸) ∧ ({𝐶, 𝐵} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) ∧ (({𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐵, 𝐴} ∈ ran 𝐸) ∧ ({𝐴, 𝐶} ∈ ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸))) ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))
222221a1i 11 . . . 4 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → (((({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸) ∧ ({𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) ∧ (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸) ∧ ({𝐶, 𝐵} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) ∧ (({𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐵, 𝐴} ∈ ran 𝐸) ∧ ({𝐴, 𝐶} ∈ ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸))) ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)))
223173, 222bitrd 267 . . 3 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → (((∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 ∧ ∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝐶}} ⊆ ran 𝐸) ∧ (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐵}, {𝑥, 𝐶}} ⊆ ran 𝐸 ∧ ∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐵}, {𝑥, 𝐴}} ⊆ ran 𝐸) ∧ (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐶}, {𝑥, 𝐴}} ⊆ ran 𝐸 ∧ ∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐶}, {𝑥, 𝐵}} ⊆ ran 𝐸)) ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)))
2247, 95, 2233bitrd 293 . 2 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → ({𝐴, 𝐵, 𝐶} FriendGrph 𝐸 ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)))
225224ex 449 1 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ({𝐴, 𝐵, 𝐶} USGrph 𝐸 → ({𝐴, 𝐵, 𝐶} FriendGrph 𝐸 ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃!wreu 2898  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  {csn 4125  {cpr 4127  {ctp 4129   class class class wbr 4583  ran crn 5039   USGrph cusg 25859   FriendGrph cfrgra 26515 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-frgra 26516 This theorem is referenced by:  3vfriswmgra  26532
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