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Theorem nbhashuvtx1 26442
 Description: If the number of the neighbors of a vertex in a finite graph is the number of vertices of the graph minus 1, each vertex except the first mentioned vertex is a neighbor of this vertex. (Contributed by Alexander van der Vekens, 14-Jul-2018.)
Assertion
Ref Expression
nbhashuvtx1 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) = ((#‘𝑉) − 1) → ((𝑀𝑉𝑀𝑁) → 𝑀 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁))))

Proof of Theorem nbhashuvtx1
StepHypRef Expression
1 ax-1 6 . . 3 (𝑀 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁) → ((𝑀𝑉𝑀𝑁) → 𝑀 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁)))
212a1d 26 . 2 (𝑀 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁) → ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) = ((#‘𝑉) − 1) → ((𝑀𝑉𝑀𝑁) → 𝑀 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁)))))
3 df-nel 2783 . . 3 (𝑀 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑁) ↔ ¬ 𝑀 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁))
4 nbgrassvwo2 25967 . . . . . . 7 ((𝑉 USGrph 𝐸𝑀 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑁)) → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) ⊆ (𝑉 ∖ {𝑀, 𝑁}))
54ex 449 . . . . . 6 (𝑉 USGrph 𝐸 → (𝑀 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑁) → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) ⊆ (𝑉 ∖ {𝑀, 𝑁})))
653ad2ant1 1075 . . . . 5 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → (𝑀 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑁) → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) ⊆ (𝑉 ∖ {𝑀, 𝑁})))
7 difexg 4735 . . . . . . 7 (𝑉 ∈ Fin → (𝑉 ∖ {𝑀, 𝑁}) ∈ V)
873ad2ant2 1076 . . . . . 6 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → (𝑉 ∖ {𝑀, 𝑁}) ∈ V)
9 hashss 13058 . . . . . . 7 (((𝑉 ∖ {𝑀, 𝑁}) ∈ V ∧ (⟨𝑉, 𝐸⟩ Neighbors 𝑁) ⊆ (𝑉 ∖ {𝑀, 𝑁})) → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) ≤ (#‘(𝑉 ∖ {𝑀, 𝑁})))
109ex 449 . . . . . 6 ((𝑉 ∖ {𝑀, 𝑁}) ∈ V → ((⟨𝑉, 𝐸⟩ Neighbors 𝑁) ⊆ (𝑉 ∖ {𝑀, 𝑁}) → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) ≤ (#‘(𝑉 ∖ {𝑀, 𝑁}))))
118, 10syl 17 . . . . 5 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → ((⟨𝑉, 𝐸⟩ Neighbors 𝑁) ⊆ (𝑉 ∖ {𝑀, 𝑁}) → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) ≤ (#‘(𝑉 ∖ {𝑀, 𝑁}))))
12 simpl2 1058 . . . . . . . . . . . 12 (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) ∧ (𝑀𝑉𝑀𝑁)) → 𝑉 ∈ Fin)
13 simpl 472 . . . . . . . . . . . . 13 ((𝑀𝑉𝑀𝑁) → 𝑀𝑉)
14 simp3 1056 . . . . . . . . . . . . 13 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → 𝑁𝑉)
15 prssi 4293 . . . . . . . . . . . . 13 ((𝑀𝑉𝑁𝑉) → {𝑀, 𝑁} ⊆ 𝑉)
1613, 14, 15syl2anr 494 . . . . . . . . . . . 12 (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) ∧ (𝑀𝑉𝑀𝑁)) → {𝑀, 𝑁} ⊆ 𝑉)
17 hashssdif 13061 . . . . . . . . . . . 12 ((𝑉 ∈ Fin ∧ {𝑀, 𝑁} ⊆ 𝑉) → (#‘(𝑉 ∖ {𝑀, 𝑁})) = ((#‘𝑉) − (#‘{𝑀, 𝑁})))
1812, 16, 17syl2anc 691 . . . . . . . . . . 11 (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) ∧ (𝑀𝑉𝑀𝑁)) → (#‘(𝑉 ∖ {𝑀, 𝑁})) = ((#‘𝑉) − (#‘{𝑀, 𝑁})))
19 simprr 792 . . . . . . . . . . . . 13 (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) ∧ (𝑀𝑉𝑀𝑁)) → 𝑀𝑁)
20 hashprgOLD 13044 . . . . . . . . . . . . . 14 ((𝑀𝑉𝑁𝑉) → (𝑀𝑁 ↔ (#‘{𝑀, 𝑁}) = 2))
2113, 14, 20syl2anr 494 . . . . . . . . . . . . 13 (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) ∧ (𝑀𝑉𝑀𝑁)) → (𝑀𝑁 ↔ (#‘{𝑀, 𝑁}) = 2))
2219, 21mpbid 221 . . . . . . . . . . . 12 (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) ∧ (𝑀𝑉𝑀𝑁)) → (#‘{𝑀, 𝑁}) = 2)
2322oveq2d 6565 . . . . . . . . . . 11 (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) ∧ (𝑀𝑉𝑀𝑁)) → ((#‘𝑉) − (#‘{𝑀, 𝑁})) = ((#‘𝑉) − 2))
2418, 23eqtrd 2644 . . . . . . . . . 10 (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) ∧ (𝑀𝑉𝑀𝑁)) → (#‘(𝑉 ∖ {𝑀, 𝑁})) = ((#‘𝑉) − 2))
2524breq2d 4595 . . . . . . . . 9 (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) ∧ (𝑀𝑉𝑀𝑁)) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) ≤ (#‘(𝑉 ∖ {𝑀, 𝑁})) ↔ (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) ≤ ((#‘𝑉) − 2)))
26 nbhashnn0 26441 . . . . . . . . . . . . . 14 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) ∈ ℕ0)
2726nn0zd 11356 . . . . . . . . . . . . 13 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) ∈ ℤ)
28 hashcl 13009 . . . . . . . . . . . . . . 15 (𝑉 ∈ Fin → (#‘𝑉) ∈ ℕ0)
29 nn0z 11277 . . . . . . . . . . . . . . 15 ((#‘𝑉) ∈ ℕ0 → (#‘𝑉) ∈ ℤ)
30 peano2zm 11297 . . . . . . . . . . . . . . 15 ((#‘𝑉) ∈ ℤ → ((#‘𝑉) − 1) ∈ ℤ)
3128, 29, 303syl 18 . . . . . . . . . . . . . 14 (𝑉 ∈ Fin → ((#‘𝑉) − 1) ∈ ℤ)
32313ad2ant2 1076 . . . . . . . . . . . . 13 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → ((#‘𝑉) − 1) ∈ ℤ)
33 zltlem1 11307 . . . . . . . . . . . . 13 (((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) ∈ ℤ ∧ ((#‘𝑉) − 1) ∈ ℤ) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) < ((#‘𝑉) − 1) ↔ (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) ≤ (((#‘𝑉) − 1) − 1)))
3427, 32, 33syl2anc 691 . . . . . . . . . . . 12 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) < ((#‘𝑉) − 1) ↔ (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) ≤ (((#‘𝑉) − 1) − 1)))
3528nn0cnd 11230 . . . . . . . . . . . . . . . 16 (𝑉 ∈ Fin → (#‘𝑉) ∈ ℂ)
36353ad2ant2 1076 . . . . . . . . . . . . . . 15 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → (#‘𝑉) ∈ ℂ)
37 1cnd 9935 . . . . . . . . . . . . . . 15 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → 1 ∈ ℂ)
3836, 37, 37subsub4d 10302 . . . . . . . . . . . . . 14 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → (((#‘𝑉) − 1) − 1) = ((#‘𝑉) − (1 + 1)))
39 1p1e2 11011 . . . . . . . . . . . . . . 15 (1 + 1) = 2
4039oveq2i 6560 . . . . . . . . . . . . . 14 ((#‘𝑉) − (1 + 1)) = ((#‘𝑉) − 2)
4138, 40syl6eq 2660 . . . . . . . . . . . . 13 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → (((#‘𝑉) − 1) − 1) = ((#‘𝑉) − 2))
4241breq2d 4595 . . . . . . . . . . . 12 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) ≤ (((#‘𝑉) − 1) − 1) ↔ (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) ≤ ((#‘𝑉) − 2)))
4334, 42bitrd 267 . . . . . . . . . . 11 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) < ((#‘𝑉) − 1) ↔ (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) ≤ ((#‘𝑉) − 2)))
4443adantr 480 . . . . . . . . . 10 (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) ∧ (𝑀𝑉𝑀𝑁)) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) < ((#‘𝑉) − 1) ↔ (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) ≤ ((#‘𝑉) − 2)))
4526nn0red 11229 . . . . . . . . . . . . . . 15 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) ∈ ℝ)
4645adantr 480 . . . . . . . . . . . . . 14 (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) ∧ (𝑀𝑉𝑀𝑁)) → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) ∈ ℝ)
4746adantr 480 . . . . . . . . . . . . 13 ((((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) ∧ (𝑀𝑉𝑀𝑁)) ∧ (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) < ((#‘𝑉) − 1)) → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) ∈ ℝ)
48 simpr 476 . . . . . . . . . . . . 13 ((((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) ∧ (𝑀𝑉𝑀𝑁)) ∧ (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) < ((#‘𝑉) − 1)) → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) < ((#‘𝑉) − 1))
4947, 48ltned 10052 . . . . . . . . . . . 12 ((((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) ∧ (𝑀𝑉𝑀𝑁)) ∧ (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) < ((#‘𝑉) − 1)) → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) ≠ ((#‘𝑉) − 1))
5049ex 449 . . . . . . . . . . 11 (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) ∧ (𝑀𝑉𝑀𝑁)) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) < ((#‘𝑉) − 1) → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) ≠ ((#‘𝑉) − 1)))
51 eqneqall 2793 . . . . . . . . . . . 12 ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) = ((#‘𝑉) − 1) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) ≠ ((#‘𝑉) − 1) → 𝑀 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁)))
5251com12 32 . . . . . . . . . . 11 ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) ≠ ((#‘𝑉) − 1) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) = ((#‘𝑉) − 1) → 𝑀 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁)))
5350, 52syl6 34 . . . . . . . . . 10 (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) ∧ (𝑀𝑉𝑀𝑁)) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) < ((#‘𝑉) − 1) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) = ((#‘𝑉) − 1) → 𝑀 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁))))
5444, 53sylbird 249 . . . . . . . . 9 (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) ∧ (𝑀𝑉𝑀𝑁)) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) ≤ ((#‘𝑉) − 2) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) = ((#‘𝑉) − 1) → 𝑀 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁))))
5525, 54sylbid 229 . . . . . . . 8 (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) ∧ (𝑀𝑉𝑀𝑁)) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) ≤ (#‘(𝑉 ∖ {𝑀, 𝑁})) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) = ((#‘𝑉) − 1) → 𝑀 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁))))
5655ex 449 . . . . . . 7 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → ((𝑀𝑉𝑀𝑁) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) ≤ (#‘(𝑉 ∖ {𝑀, 𝑁})) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) = ((#‘𝑉) − 1) → 𝑀 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁)))))
5756com23 84 . . . . . 6 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) ≤ (#‘(𝑉 ∖ {𝑀, 𝑁})) → ((𝑀𝑉𝑀𝑁) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) = ((#‘𝑉) − 1) → 𝑀 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁)))))
5857com34 89 . . . . 5 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) ≤ (#‘(𝑉 ∖ {𝑀, 𝑁})) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) = ((#‘𝑉) − 1) → ((𝑀𝑉𝑀𝑁) → 𝑀 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁)))))
596, 11, 583syld 58 . . . 4 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → (𝑀 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑁) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) = ((#‘𝑉) − 1) → ((𝑀𝑉𝑀𝑁) → 𝑀 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁)))))
6059com12 32 . . 3 (𝑀 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑁) → ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) = ((#‘𝑉) − 1) → ((𝑀𝑉𝑀𝑁) → 𝑀 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁)))))
613, 60sylbir 224 . 2 𝑀 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁) → ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) = ((#‘𝑉) − 1) → ((𝑀𝑉𝑀𝑁) → 𝑀 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁)))))
622, 61pm2.61i 175 1 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) = ((#‘𝑉) − 1) → ((𝑀𝑉𝑀𝑁) → 𝑀 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   ∉ wnel 2781  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  {cpr 4127  ⟨cop 4131   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  Fincfn 7841  ℂcc 9813  ℝcr 9814  1c1 9816   + caddc 9818   < clt 9953   ≤ cle 9954   − cmin 10145  2c2 10947  ℕ0cn0 11169  ℤcz 11254  #chash 12979   USGrph cusg 25859   Neighbors cnbgra 25946 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-2o 7448  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-xnn0 11241  df-z 11255  df-uz 11564  df-xadd 11823  df-fz 12198  df-hash 12980  df-usgra 25862  df-nbgra 25949  df-vdgr 26421 This theorem is referenced by:  nbhashuvtx  26443
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