Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  nbcusgra Structured version   Visualization version   GIF version

Theorem nbcusgra 25992
 Description: In a complete (undirected simple) graph, each vertex has all other vertices as neighbors. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
Assertion
Ref Expression
nbcusgra ((𝑉 ComplUSGrph 𝐸𝑁𝑉) → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = (𝑉 ∖ {𝑁}))

Proof of Theorem nbcusgra
Dummy variables 𝑘 𝑛 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cusisusgra 25987 . . 3 (𝑉 ComplUSGrph 𝐸𝑉 USGrph 𝐸)
2 nbusgra 25957 . . . . . . 7 (𝑉 USGrph 𝐸 → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸})
32adantr 480 . . . . . 6 ((𝑉 USGrph 𝐸𝑉 ComplUSGrph 𝐸) → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸})
43adantr 480 . . . . 5 (((𝑉 USGrph 𝐸𝑉 ComplUSGrph 𝐸) ∧ 𝑁𝑉) → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸})
5 usgrav 25867 . . . . . . . . . 10 (𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
6 iscusgra 25985 . . . . . . . . . 10 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉 ComplUSGrph 𝐸 ↔ (𝑉 USGrph 𝐸 ∧ ∀𝑥𝑉𝑘 ∈ (𝑉 ∖ {𝑥}){𝑘, 𝑥} ∈ ran 𝐸)))
75, 6syl 17 . . . . . . . . 9 (𝑉 USGrph 𝐸 → (𝑉 ComplUSGrph 𝐸 ↔ (𝑉 USGrph 𝐸 ∧ ∀𝑥𝑉𝑘 ∈ (𝑉 ∖ {𝑥}){𝑘, 𝑥} ∈ ran 𝐸)))
87biimpa 500 . . . . . . . 8 ((𝑉 USGrph 𝐸𝑉 ComplUSGrph 𝐸) → (𝑉 USGrph 𝐸 ∧ ∀𝑥𝑉𝑘 ∈ (𝑉 ∖ {𝑥}){𝑘, 𝑥} ∈ ran 𝐸))
9 df-ral 2901 . . . . . . . . . 10 (∀𝑥𝑉𝑘 ∈ (𝑉 ∖ {𝑥}){𝑘, 𝑥} ∈ ran 𝐸 ↔ ∀𝑥(𝑥𝑉 → ∀𝑘 ∈ (𝑉 ∖ {𝑥}){𝑘, 𝑥} ∈ ran 𝐸))
10 preq2 4213 . . . . . . . . . . . . . . 15 (𝑛 = 𝑥 → {𝑁, 𝑛} = {𝑁, 𝑥})
1110eleq1d 2672 . . . . . . . . . . . . . 14 (𝑛 = 𝑥 → ({𝑁, 𝑛} ∈ ran 𝐸 ↔ {𝑁, 𝑥} ∈ ran 𝐸))
1211elrab 3331 . . . . . . . . . . . . 13 (𝑥 ∈ {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸} ↔ (𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸))
13 pm2.24 120 . . . . . . . . . . . . . . . . . . 19 (𝑥𝑉 → (¬ 𝑥𝑉𝑥 ∈ (𝑉 ∖ {𝑁})))
1413adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) → (¬ 𝑥𝑉𝑥 ∈ (𝑉 ∖ {𝑁})))
1514com12 32 . . . . . . . . . . . . . . . . 17 𝑥𝑉 → ((𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) → 𝑥 ∈ (𝑉 ∖ {𝑁})))
16 eldif 3550 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ (𝑉 ∖ {𝑁}) ↔ (𝑥𝑉 ∧ ¬ 𝑥 ∈ {𝑁}))
17 pm2.24 120 . . . . . . . . . . . . . . . . . . . 20 (𝑥𝑉 → (¬ 𝑥𝑉 → (𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸)))
1817adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝑥𝑉 ∧ ¬ 𝑥 ∈ {𝑁}) → (¬ 𝑥𝑉 → (𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸)))
1916, 18sylbi 206 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (𝑉 ∖ {𝑁}) → (¬ 𝑥𝑉 → (𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸)))
2019com12 32 . . . . . . . . . . . . . . . . 17 𝑥𝑉 → (𝑥 ∈ (𝑉 ∖ {𝑁}) → (𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸)))
2115, 20impbid 201 . . . . . . . . . . . . . . . 16 𝑥𝑉 → ((𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ↔ 𝑥 ∈ (𝑉 ∖ {𝑁})))
2221a1d 25 . . . . . . . . . . . . . . 15 𝑥𝑉 → ((𝑉 USGrph 𝐸𝑁𝑉) → ((𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ↔ 𝑥 ∈ (𝑉 ∖ {𝑁}))))
23 usgraedgrn 25910 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑉 USGrph 𝐸 ∧ {𝑁, 𝑥} ∈ ran 𝐸) → 𝑁𝑥)
24 elsni 4142 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 ∈ {𝑁} → 𝑥 = 𝑁)
25 eqcom 2617 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑁 = 𝑥𝑥 = 𝑁)
2624, 25sylibr 223 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 ∈ {𝑁} → 𝑁 = 𝑥)
2726necon3ai 2807 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑁𝑥 → ¬ 𝑥 ∈ {𝑁})
2823, 27syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑉 USGrph 𝐸 ∧ {𝑁, 𝑥} ∈ ran 𝐸) → ¬ 𝑥 ∈ {𝑁})
2928ex 449 . . . . . . . . . . . . . . . . . . . . 21 (𝑉 USGrph 𝐸 → ({𝑁, 𝑥} ∈ ran 𝐸 → ¬ 𝑥 ∈ {𝑁}))
3029ad2antrl 760 . . . . . . . . . . . . . . . . . . . 20 ((∀𝑘 ∈ (𝑉 ∖ {𝑥}){𝑘, 𝑥} ∈ ran 𝐸 ∧ (𝑉 USGrph 𝐸𝑁𝑉)) → ({𝑁, 𝑥} ∈ ran 𝐸 → ¬ 𝑥 ∈ {𝑁}))
3130adantr 480 . . . . . . . . . . . . . . . . . . 19 (((∀𝑘 ∈ (𝑉 ∖ {𝑥}){𝑘, 𝑥} ∈ ran 𝐸 ∧ (𝑉 USGrph 𝐸𝑁𝑉)) ∧ 𝑥𝑉) → ({𝑁, 𝑥} ∈ ran 𝐸 → ¬ 𝑥 ∈ {𝑁}))
32 simpr 476 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((¬ 𝑥 ∈ {𝑁} ∧ 𝑁𝑉) → 𝑁𝑉)
33 elsng 4139 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑁𝑉 → (𝑁 ∈ {𝑥} ↔ 𝑁 = 𝑥))
34 velsn 4141 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑥 ∈ {𝑁} ↔ 𝑥 = 𝑁)
3525, 34sylbb2 227 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑁 = 𝑥𝑥 ∈ {𝑁})
3633, 35syl6bi 242 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑁𝑉 → (𝑁 ∈ {𝑥} → 𝑥 ∈ {𝑁}))
3736con3d 147 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑁𝑉 → (¬ 𝑥 ∈ {𝑁} → ¬ 𝑁 ∈ {𝑥}))
3837impcom 445 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((¬ 𝑥 ∈ {𝑁} ∧ 𝑁𝑉) → ¬ 𝑁 ∈ {𝑥})
3932, 38eldifd 3551 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((¬ 𝑥 ∈ {𝑁} ∧ 𝑁𝑉) → 𝑁 ∈ (𝑉 ∖ {𝑥}))
40 preq1 4212 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑘 = 𝑁 → {𝑘, 𝑥} = {𝑁, 𝑥})
4140eleq1d 2672 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑘 = 𝑁 → ({𝑘, 𝑥} ∈ ran 𝐸 ↔ {𝑁, 𝑥} ∈ ran 𝐸))
4241rspcva 3280 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑁 ∈ (𝑉 ∖ {𝑥}) ∧ ∀𝑘 ∈ (𝑉 ∖ {𝑥}){𝑘, 𝑥} ∈ ran 𝐸) → {𝑁, 𝑥} ∈ ran 𝐸)
43422a1d 26 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑁 ∈ (𝑉 ∖ {𝑥}) ∧ ∀𝑘 ∈ (𝑉 ∖ {𝑥}){𝑘, 𝑥} ∈ ran 𝐸) → (𝑥𝑉 → (𝑉 USGrph 𝐸 → {𝑁, 𝑥} ∈ ran 𝐸)))
4443ex 449 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑁 ∈ (𝑉 ∖ {𝑥}) → (∀𝑘 ∈ (𝑉 ∖ {𝑥}){𝑘, 𝑥} ∈ ran 𝐸 → (𝑥𝑉 → (𝑉 USGrph 𝐸 → {𝑁, 𝑥} ∈ ran 𝐸))))
4544com24 93 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑁 ∈ (𝑉 ∖ {𝑥}) → (𝑉 USGrph 𝐸 → (𝑥𝑉 → (∀𝑘 ∈ (𝑉 ∖ {𝑥}){𝑘, 𝑥} ∈ ran 𝐸 → {𝑁, 𝑥} ∈ ran 𝐸))))
4639, 45syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((¬ 𝑥 ∈ {𝑁} ∧ 𝑁𝑉) → (𝑉 USGrph 𝐸 → (𝑥𝑉 → (∀𝑘 ∈ (𝑉 ∖ {𝑥}){𝑘, 𝑥} ∈ ran 𝐸 → {𝑁, 𝑥} ∈ ran 𝐸))))
4746ex 449 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑥 ∈ {𝑁} → (𝑁𝑉 → (𝑉 USGrph 𝐸 → (𝑥𝑉 → (∀𝑘 ∈ (𝑉 ∖ {𝑥}){𝑘, 𝑥} ∈ ran 𝐸 → {𝑁, 𝑥} ∈ ran 𝐸)))))
4847com13 86 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑉 USGrph 𝐸 → (𝑁𝑉 → (¬ 𝑥 ∈ {𝑁} → (𝑥𝑉 → (∀𝑘 ∈ (𝑉 ∖ {𝑥}){𝑘, 𝑥} ∈ ran 𝐸 → {𝑁, 𝑥} ∈ ran 𝐸)))))
4948imp 444 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑉 USGrph 𝐸𝑁𝑉) → (¬ 𝑥 ∈ {𝑁} → (𝑥𝑉 → (∀𝑘 ∈ (𝑉 ∖ {𝑥}){𝑘, 𝑥} ∈ ran 𝐸 → {𝑁, 𝑥} ∈ ran 𝐸))))
5049com12 32 . . . . . . . . . . . . . . . . . . . . 21 𝑥 ∈ {𝑁} → ((𝑉 USGrph 𝐸𝑁𝑉) → (𝑥𝑉 → (∀𝑘 ∈ (𝑉 ∖ {𝑥}){𝑘, 𝑥} ∈ ran 𝐸 → {𝑁, 𝑥} ∈ ran 𝐸))))
5150com14 94 . . . . . . . . . . . . . . . . . . . 20 (∀𝑘 ∈ (𝑉 ∖ {𝑥}){𝑘, 𝑥} ∈ ran 𝐸 → ((𝑉 USGrph 𝐸𝑁𝑉) → (𝑥𝑉 → (¬ 𝑥 ∈ {𝑁} → {𝑁, 𝑥} ∈ ran 𝐸))))
5251imp31 447 . . . . . . . . . . . . . . . . . . 19 (((∀𝑘 ∈ (𝑉 ∖ {𝑥}){𝑘, 𝑥} ∈ ran 𝐸 ∧ (𝑉 USGrph 𝐸𝑁𝑉)) ∧ 𝑥𝑉) → (¬ 𝑥 ∈ {𝑁} → {𝑁, 𝑥} ∈ ran 𝐸))
5331, 52impbid 201 . . . . . . . . . . . . . . . . . 18 (((∀𝑘 ∈ (𝑉 ∖ {𝑥}){𝑘, 𝑥} ∈ ran 𝐸 ∧ (𝑉 USGrph 𝐸𝑁𝑉)) ∧ 𝑥𝑉) → ({𝑁, 𝑥} ∈ ran 𝐸 ↔ ¬ 𝑥 ∈ {𝑁}))
5453pm5.32da 671 . . . . . . . . . . . . . . . . 17 ((∀𝑘 ∈ (𝑉 ∖ {𝑥}){𝑘, 𝑥} ∈ ran 𝐸 ∧ (𝑉 USGrph 𝐸𝑁𝑉)) → ((𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ↔ (𝑥𝑉 ∧ ¬ 𝑥 ∈ {𝑁})))
5554, 16syl6bbr 277 . . . . . . . . . . . . . . . 16 ((∀𝑘 ∈ (𝑉 ∖ {𝑥}){𝑘, 𝑥} ∈ ran 𝐸 ∧ (𝑉 USGrph 𝐸𝑁𝑉)) → ((𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ↔ 𝑥 ∈ (𝑉 ∖ {𝑁})))
5655ex 449 . . . . . . . . . . . . . . 15 (∀𝑘 ∈ (𝑉 ∖ {𝑥}){𝑘, 𝑥} ∈ ran 𝐸 → ((𝑉 USGrph 𝐸𝑁𝑉) → ((𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ↔ 𝑥 ∈ (𝑉 ∖ {𝑁}))))
5722, 56ja 172 . . . . . . . . . . . . . 14 ((𝑥𝑉 → ∀𝑘 ∈ (𝑉 ∖ {𝑥}){𝑘, 𝑥} ∈ ran 𝐸) → ((𝑉 USGrph 𝐸𝑁𝑉) → ((𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ↔ 𝑥 ∈ (𝑉 ∖ {𝑁}))))
5857impcom 445 . . . . . . . . . . . . 13 (((𝑉 USGrph 𝐸𝑁𝑉) ∧ (𝑥𝑉 → ∀𝑘 ∈ (𝑉 ∖ {𝑥}){𝑘, 𝑥} ∈ ran 𝐸)) → ((𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ↔ 𝑥 ∈ (𝑉 ∖ {𝑁})))
5912, 58syl5bb 271 . . . . . . . . . . . 12 (((𝑉 USGrph 𝐸𝑁𝑉) ∧ (𝑥𝑉 → ∀𝑘 ∈ (𝑉 ∖ {𝑥}){𝑘, 𝑥} ∈ ran 𝐸)) → (𝑥 ∈ {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸} ↔ 𝑥 ∈ (𝑉 ∖ {𝑁})))
6059ex 449 . . . . . . . . . . 11 ((𝑉 USGrph 𝐸𝑁𝑉) → ((𝑥𝑉 → ∀𝑘 ∈ (𝑉 ∖ {𝑥}){𝑘, 𝑥} ∈ ran 𝐸) → (𝑥 ∈ {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸} ↔ 𝑥 ∈ (𝑉 ∖ {𝑁}))))
6160alimdv 1832 . . . . . . . . . 10 ((𝑉 USGrph 𝐸𝑁𝑉) → (∀𝑥(𝑥𝑉 → ∀𝑘 ∈ (𝑉 ∖ {𝑥}){𝑘, 𝑥} ∈ ran 𝐸) → ∀𝑥(𝑥 ∈ {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸} ↔ 𝑥 ∈ (𝑉 ∖ {𝑁}))))
629, 61syl5bi 231 . . . . . . . . 9 ((𝑉 USGrph 𝐸𝑁𝑉) → (∀𝑥𝑉𝑘 ∈ (𝑉 ∖ {𝑥}){𝑘, 𝑥} ∈ ran 𝐸 → ∀𝑥(𝑥 ∈ {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸} ↔ 𝑥 ∈ (𝑉 ∖ {𝑁}))))
6362impancom 455 . . . . . . . 8 ((𝑉 USGrph 𝐸 ∧ ∀𝑥𝑉𝑘 ∈ (𝑉 ∖ {𝑥}){𝑘, 𝑥} ∈ ran 𝐸) → (𝑁𝑉 → ∀𝑥(𝑥 ∈ {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸} ↔ 𝑥 ∈ (𝑉 ∖ {𝑁}))))
648, 63syl 17 . . . . . . 7 ((𝑉 USGrph 𝐸𝑉 ComplUSGrph 𝐸) → (𝑁𝑉 → ∀𝑥(𝑥 ∈ {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸} ↔ 𝑥 ∈ (𝑉 ∖ {𝑁}))))
6564imp 444 . . . . . 6 (((𝑉 USGrph 𝐸𝑉 ComplUSGrph 𝐸) ∧ 𝑁𝑉) → ∀𝑥(𝑥 ∈ {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸} ↔ 𝑥 ∈ (𝑉 ∖ {𝑁})))
66 dfcleq 2604 . . . . . 6 ({𝑛𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸} = (𝑉 ∖ {𝑁}) ↔ ∀𝑥(𝑥 ∈ {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸} ↔ 𝑥 ∈ (𝑉 ∖ {𝑁})))
6765, 66sylibr 223 . . . . 5 (((𝑉 USGrph 𝐸𝑉 ComplUSGrph 𝐸) ∧ 𝑁𝑉) → {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸} = (𝑉 ∖ {𝑁}))
684, 67eqtrd 2644 . . . 4 (((𝑉 USGrph 𝐸𝑉 ComplUSGrph 𝐸) ∧ 𝑁𝑉) → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = (𝑉 ∖ {𝑁}))
6968ex 449 . . 3 ((𝑉 USGrph 𝐸𝑉 ComplUSGrph 𝐸) → (𝑁𝑉 → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = (𝑉 ∖ {𝑁})))
701, 69mpancom 700 . 2 (𝑉 ComplUSGrph 𝐸 → (𝑁𝑉 → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = (𝑉 ∖ {𝑁})))
7170imp 444 1 ((𝑉 ComplUSGrph 𝐸𝑁𝑉) → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = (𝑉 ∖ {𝑁}))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  {crab 2900  Vcvv 3173   ∖ cdif 3537  {csn 4125  {cpr 4127  ⟨cop 4131   class class class wbr 4583  ran crn 5039  (class class class)co 6549   USGrph cusg 25859   Neighbors cnbgra 25946   ComplUSGrph ccusgra 25947 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-nbgra 25949  df-cusgra 25950 This theorem is referenced by:  cusgrasizeindslem2  26003  cusgraisrusgra  26465
 Copyright terms: Public domain W3C validator