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Theorem vdn1frgrav2 26314
Description: Any vertex in a friendship graph does not have degree 1, see remark 2 in [MertziosUnger] p. 153 (after Proposition 1): "... no node v of it [a friendship graph] may have deg(v) = 1.". (Contributed by Alexander van der Vekens, 10-Dec-2017.)
Assertion
Ref Expression
vdn1frgrav2 ((𝑉 FriendGrph 𝐸𝐸 ∈ Fin ∧ 𝑁𝑉) → (1 < (#‘𝑉) → ((𝑉 VDeg 𝐸)‘𝑁) ≠ 1))

Proof of Theorem vdn1frgrav2
Dummy variables 𝑎 𝑏 𝑐 𝑥 𝑖 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frisusgra 26281 . . . . . . 7 (𝑉 FriendGrph 𝐸𝑉 USGrph 𝐸)
213ad2ant1 1074 . . . . . 6 ((𝑉 FriendGrph 𝐸𝐸 ∈ Fin ∧ 𝑁𝑉) → 𝑉 USGrph 𝐸)
3 simp3 1055 . . . . . 6 ((𝑉 FriendGrph 𝐸𝐸 ∈ Fin ∧ 𝑁𝑉) → 𝑁𝑉)
4 simp2 1054 . . . . . 6 ((𝑉 FriendGrph 𝐸𝐸 ∈ Fin ∧ 𝑁𝑉) → 𝐸 ∈ Fin)
52, 3, 43jca 1234 . . . . 5 ((𝑉 FriendGrph 𝐸𝐸 ∈ Fin ∧ 𝑁𝑉) → (𝑉 USGrph 𝐸𝑁𝑉𝐸 ∈ Fin))
65adantr 479 . . . 4 (((𝑉 FriendGrph 𝐸𝐸 ∈ Fin ∧ 𝑁𝑉) ∧ 1 < (#‘𝑉)) → (𝑉 USGrph 𝐸𝑁𝑉𝐸 ∈ Fin))
7 vdusgraval 26196 . . . . 5 ((𝑉 USGrph 𝐸𝑁𝑉) → ((𝑉 VDeg 𝐸)‘𝑁) = (#‘{𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}))
873adant3 1073 . . . 4 ((𝑉 USGrph 𝐸𝑁𝑉𝐸 ∈ Fin) → ((𝑉 VDeg 𝐸)‘𝑁) = (#‘{𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}))
96, 8syl 17 . . 3 (((𝑉 FriendGrph 𝐸𝐸 ∈ Fin ∧ 𝑁𝑉) ∧ 1 < (#‘𝑉)) → ((𝑉 VDeg 𝐸)‘𝑁) = (#‘{𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}))
10 3cyclfrgrarn2 26303 . . . . . 6 ((𝑉 FriendGrph 𝐸 ∧ 1 < (#‘𝑉)) → ∀𝑎𝑉𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))
11103ad2antl1 1215 . . . . 5 (((𝑉 FriendGrph 𝐸𝐸 ∈ Fin ∧ 𝑁𝑉) ∧ 1 < (#‘𝑉)) → ∀𝑎𝑉𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))
12 preq1 4207 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑁 → {𝑎, 𝑏} = {𝑁, 𝑏})
1312eleq1d 2667 . . . . . . . . . . . . . . 15 (𝑎 = 𝑁 → ({𝑎, 𝑏} ∈ ran 𝐸 ↔ {𝑁, 𝑏} ∈ ran 𝐸))
14 preq2 4208 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑁 → {𝑐, 𝑎} = {𝑐, 𝑁})
1514eleq1d 2667 . . . . . . . . . . . . . . 15 (𝑎 = 𝑁 → ({𝑐, 𝑎} ∈ ran 𝐸 ↔ {𝑐, 𝑁} ∈ ran 𝐸))
1613, 153anbi13d 1392 . . . . . . . . . . . . . 14 (𝑎 = 𝑁 → (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸) ↔ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)))
1716anbi2d 735 . . . . . . . . . . . . 13 (𝑎 = 𝑁 → ((𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) ↔ (𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸))))
18172rexbidv 3034 . . . . . . . . . . . 12 (𝑎 = 𝑁 → (∃𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) ↔ ∃𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸))))
1918rspcva 3275 . . . . . . . . . . 11 ((𝑁𝑉 ∧ ∀𝑎𝑉𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))) → ∃𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)))
201adantl 480 . . . . . . . . . . . . . . . . 17 ((((𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)) ∧ 𝑁𝑉) ∧ 𝑉 FriendGrph 𝐸) → 𝑉 USGrph 𝐸)
21 simplr 787 . . . . . . . . . . . . . . . . 17 ((((𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)) ∧ 𝑁𝑉) ∧ 𝑉 FriendGrph 𝐸) → 𝑁𝑉)
22 simplll 793 . . . . . . . . . . . . . . . . 17 ((((𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)) ∧ 𝑁𝑉) ∧ 𝑉 FriendGrph 𝐸) → 𝑏𝑐)
23 3simpb 1051 . . . . . . . . . . . . . . . . . 18 (({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸) → ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸))
2423ad3antlr 762 . . . . . . . . . . . . . . . . 17 ((((𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)) ∧ 𝑁𝑉) ∧ 𝑉 FriendGrph 𝐸) → ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸))
25 usgra2edg1 25674 . . . . . . . . . . . . . . . . 17 (((𝑉 USGrph 𝐸𝑁𝑉𝑏𝑐) ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)) → ¬ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥))
2620, 21, 22, 24, 25syl31anc 1320 . . . . . . . . . . . . . . . 16 ((((𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)) ∧ 𝑁𝑉) ∧ 𝑉 FriendGrph 𝐸) → ¬ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥))
27262a1d 26 . . . . . . . . . . . . . . 15 ((((𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)) ∧ 𝑁𝑉) ∧ 𝑉 FriendGrph 𝐸) → (1 < (#‘𝑉) → (𝐸 ∈ Fin → ¬ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥))))
2827ex 448 . . . . . . . . . . . . . 14 (((𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)) ∧ 𝑁𝑉) → (𝑉 FriendGrph 𝐸 → (1 < (#‘𝑉) → (𝐸 ∈ Fin → ¬ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥)))))
2928ex 448 . . . . . . . . . . . . 13 ((𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)) → (𝑁𝑉 → (𝑉 FriendGrph 𝐸 → (1 < (#‘𝑉) → (𝐸 ∈ Fin → ¬ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥))))))
3029a1i 11 . . . . . . . . . . . 12 ((𝑏𝑉𝑐𝑉) → ((𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)) → (𝑁𝑉 → (𝑉 FriendGrph 𝐸 → (1 < (#‘𝑉) → (𝐸 ∈ Fin → ¬ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥)))))))
3130rexlimivv 3013 . . . . . . . . . . 11 (∃𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)) → (𝑁𝑉 → (𝑉 FriendGrph 𝐸 → (1 < (#‘𝑉) → (𝐸 ∈ Fin → ¬ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥))))))
3219, 31syl 17 . . . . . . . . . 10 ((𝑁𝑉 ∧ ∀𝑎𝑉𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))) → (𝑁𝑉 → (𝑉 FriendGrph 𝐸 → (1 < (#‘𝑉) → (𝐸 ∈ Fin → ¬ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥))))))
3332ex 448 . . . . . . . . 9 (𝑁𝑉 → (∀𝑎𝑉𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) → (𝑁𝑉 → (𝑉 FriendGrph 𝐸 → (1 < (#‘𝑉) → (𝐸 ∈ Fin → ¬ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥)))))))
3433pm2.43a 51 . . . . . . . 8 (𝑁𝑉 → (∀𝑎𝑉𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) → (𝑉 FriendGrph 𝐸 → (1 < (#‘𝑉) → (𝐸 ∈ Fin → ¬ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥))))))
3534com25 96 . . . . . . 7 (𝑁𝑉 → (𝐸 ∈ Fin → (𝑉 FriendGrph 𝐸 → (1 < (#‘𝑉) → (∀𝑎𝑉𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) → ¬ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥))))))
3635com13 85 . . . . . 6 (𝑉 FriendGrph 𝐸 → (𝐸 ∈ Fin → (𝑁𝑉 → (1 < (#‘𝑉) → (∀𝑎𝑉𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) → ¬ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥))))))
37363imp1 1271 . . . . 5 (((𝑉 FriendGrph 𝐸𝐸 ∈ Fin ∧ 𝑁𝑉) ∧ 1 < (#‘𝑉)) → (∀𝑎𝑉𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) → ¬ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥)))
3811, 37mpd 15 . . . 4 (((𝑉 FriendGrph 𝐸𝐸 ∈ Fin ∧ 𝑁𝑉) ∧ 1 < (#‘𝑉)) → ¬ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥))
39 dmfi 8102 . . . . . . . . . 10 (𝐸 ∈ Fin → dom 𝐸 ∈ Fin)
40393ad2ant2 1075 . . . . . . . . 9 ((𝑉 FriendGrph 𝐸𝐸 ∈ Fin ∧ 𝑁𝑉) → dom 𝐸 ∈ Fin)
4140adantr 479 . . . . . . . 8 (((𝑉 FriendGrph 𝐸𝐸 ∈ Fin ∧ 𝑁𝑉) ∧ 1 < (#‘𝑉)) → dom 𝐸 ∈ Fin)
42 rabexg 4730 . . . . . . . 8 (dom 𝐸 ∈ Fin → {𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)} ∈ V)
4341, 42syl 17 . . . . . . 7 (((𝑉 FriendGrph 𝐸𝐸 ∈ Fin ∧ 𝑁𝑉) ∧ 1 < (#‘𝑉)) → {𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)} ∈ V)
44 hash1snb 13016 . . . . . . 7 ({𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)} ∈ V → ((#‘{𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}) = 1 ↔ ∃𝑖{𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)} = {𝑖}))
4543, 44syl 17 . . . . . 6 (((𝑉 FriendGrph 𝐸𝐸 ∈ Fin ∧ 𝑁𝑉) ∧ 1 < (#‘𝑉)) → ((#‘{𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}) = 1 ↔ ∃𝑖{𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)} = {𝑖}))
46 reusn 4201 . . . . . 6 (∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥) ↔ ∃𝑖{𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)} = {𝑖})
4745, 46syl6bbr 276 . . . . 5 (((𝑉 FriendGrph 𝐸𝐸 ∈ Fin ∧ 𝑁𝑉) ∧ 1 < (#‘𝑉)) → ((#‘{𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}) = 1 ↔ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥)))
4847necon3abid 2813 . . . 4 (((𝑉 FriendGrph 𝐸𝐸 ∈ Fin ∧ 𝑁𝑉) ∧ 1 < (#‘𝑉)) → ((#‘{𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}) ≠ 1 ↔ ¬ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥)))
4938, 48mpbird 245 . . 3 (((𝑉 FriendGrph 𝐸𝐸 ∈ Fin ∧ 𝑁𝑉) ∧ 1 < (#‘𝑉)) → (#‘{𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}) ≠ 1)
509, 49eqnetrd 2844 . 2 (((𝑉 FriendGrph 𝐸𝐸 ∈ Fin ∧ 𝑁𝑉) ∧ 1 < (#‘𝑉)) → ((𝑉 VDeg 𝐸)‘𝑁) ≠ 1)
5150ex 448 1 ((𝑉 FriendGrph 𝐸𝐸 ∈ Fin ∧ 𝑁𝑉) → (1 < (#‘𝑉) → ((𝑉 VDeg 𝐸)‘𝑁) ≠ 1))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wex 1694  wcel 1975  wne 2775  wral 2891  wrex 2892  ∃!wreu 2893  {crab 2895  Vcvv 3168  {csn 4120  {cpr 4122   class class class wbr 4573  dom cdm 5024  ran crn 5025  cfv 5786  (class class class)co 6523  Fincfn 7814  1c1 9789   < clt 9926  #chash 12930   USGrph cusg 25621   VDeg cvdg 26182   FriendGrph cfrgra 26277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820  ax-cnex 9844  ax-resscn 9845  ax-1cn 9846  ax-icn 9847  ax-addcl 9848  ax-addrcl 9849  ax-mulcl 9850  ax-mulrcl 9851  ax-mulcom 9852  ax-addass 9853  ax-mulass 9854  ax-distr 9855  ax-i2m1 9856  ax-1ne0 9857  ax-1rid 9858  ax-rnegex 9859  ax-rrecex 9860  ax-cnre 9861  ax-pre-lttri 9862  ax-pre-lttrn 9863  ax-pre-ltadd 9864  ax-pre-mulgt0 9865
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-nel 2778  df-ral 2896  df-rex 2897  df-reu 2898  df-rmo 2899  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-pss 3551  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-tp 4125  df-op 4127  df-uni 4363  df-int 4401  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-tr 4671  df-eprel 4935  df-id 4939  df-po 4945  df-so 4946  df-fr 4983  df-we 4985  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-pred 5579  df-ord 5625  df-on 5626  df-lim 5627  df-suc 5628  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-riota 6485  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-om 6931  df-1st 7032  df-2nd 7033  df-wrecs 7267  df-recs 7328  df-rdg 7366  df-1o 7420  df-oadd 7424  df-er 7602  df-en 7815  df-dom 7816  df-sdom 7817  df-fin 7818  df-card 8621  df-cda 8846  df-pnf 9928  df-mnf 9929  df-xr 9930  df-ltxr 9931  df-le 9932  df-sub 10115  df-neg 10116  df-nn 10864  df-2 10922  df-n0 11136  df-z 11207  df-uz 11516  df-xadd 11775  df-fz 12149  df-hash 12931  df-usgra 25624  df-vdgr 26183  df-frgra 26278
This theorem is referenced by: (None)
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