nbusgrvtxm1 - Metamath Proof Explorer

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Theorem nbusgrvtxm1 28636

Description: If the number of neighbors of a vertex in a finite simple 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.) (Revised by AV, 16-Dec-2020.)

**Hypothesis**
Ref	Expression
hashnbusgrnn0.v	⊢ 𝑉 = (Vtx‘𝐺)

**Assertion**
Ref	Expression
nbusgrvtxm1	⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉) → ((♯‘(𝐺 NeighbVtx 𝑈)) = ((♯‘𝑉) − 1) → ((𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈) → 𝑀 ∈ (𝐺 NeighbVtx 𝑈))))

**Proof of Theorem nbusgrvtxm1**
Step	Hyp	Ref	Expression
1		ax-1 6	. . 3 ⊢ (𝑀 ∈ (𝐺 NeighbVtx 𝑈) → ((𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈) → 𝑀 ∈ (𝐺 NeighbVtx 𝑈)))
2	1	2a1d 26	. 2 ⊢ (𝑀 ∈ (𝐺 NeighbVtx 𝑈) → ((𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉) → ((♯‘(𝐺 NeighbVtx 𝑈)) = ((♯‘𝑉) − 1) → ((𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈) → 𝑀 ∈ (𝐺 NeighbVtx 𝑈)))))
3		simpr 486	. . . . . . . 8 ⊢ ((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉)) → (𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉))
4	3	adantr 482	. . . . . . 7 ⊢ (((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉)) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈)) → (𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉))
5		simprl 770	. . . . . . 7 ⊢ (((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉)) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈)) → 𝑀 ∈ 𝑉)
6		simpr 486	. . . . . . . 8 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈) → 𝑀 ≠ 𝑈)
7	6	adantl 483	. . . . . . 7 ⊢ (((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉)) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈)) → 𝑀 ≠ 𝑈)
8		df-nel 3048	. . . . . . . . . 10 ⊢ (𝑀 ∉ (𝐺 NeighbVtx 𝑈) ↔ ¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈))
9	8	biimpri 227	. . . . . . . . 9 ⊢ (¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) → 𝑀 ∉ (𝐺 NeighbVtx 𝑈))
10	9	adantr 482	. . . . . . . 8 ⊢ ((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉)) → 𝑀 ∉ (𝐺 NeighbVtx 𝑈))
11	10	adantr 482	. . . . . . 7 ⊢ (((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉)) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈)) → 𝑀 ∉ (𝐺 NeighbVtx 𝑈))
12		hashnbusgrnn0.v	. . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺)
13	12	nbfusgrlevtxm2 28635	. . . . . . 7 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈 ∧ 𝑀 ∉ (𝐺 NeighbVtx 𝑈))) → (♯‘(𝐺 NeighbVtx 𝑈)) ≤ ((♯‘𝑉) − 2))
14	4, 5, 7, 11, 13	syl13anc 1373	. . . . . 6 ⊢ (((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉)) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈)) → (♯‘(𝐺 NeighbVtx 𝑈)) ≤ ((♯‘𝑉) − 2))
15		breq1 5152	. . . . . . . . 9 ⊢ ((♯‘(𝐺 NeighbVtx 𝑈)) = ((♯‘𝑉) − 1) → ((♯‘(𝐺 NeighbVtx 𝑈)) ≤ ((♯‘𝑉) − 2) ↔ ((♯‘𝑉) − 1) ≤ ((♯‘𝑉) − 2)))
16	15	adantl 483	. . . . . . . 8 ⊢ ((((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉)) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈)) ∧ (♯‘(𝐺 NeighbVtx 𝑈)) = ((♯‘𝑉) − 1)) → ((♯‘(𝐺 NeighbVtx 𝑈)) ≤ ((♯‘𝑉) − 2) ↔ ((♯‘𝑉) − 1) ≤ ((♯‘𝑉) − 2)))
17	12	fusgrvtxfi 28576	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ FinUSGraph → 𝑉 ∈ Fin)
18		hashcl 14316	. . . . . . . . . . . 12 ⊢ (𝑉 ∈ Fin → (♯‘𝑉) ∈ ℕ₀)
19		nn0re 12481	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑉) ∈ ℕ₀ → (♯‘𝑉) ∈ ℝ)
20		1red 11215	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑉) ∈ ℝ → 1 ∈ ℝ)
21		2re 12286	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℝ
22	21	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑉) ∈ ℝ → 2 ∈ ℝ)
23		id 22	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑉) ∈ ℝ → (♯‘𝑉) ∈ ℝ)
24		1lt2 12383	. . . . . . . . . . . . . . . 16 ⊢ 1 < 2
25	24	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑉) ∈ ℝ → 1 < 2)
26	20, 22, 23, 25	ltsub2dd 11827	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑉) ∈ ℝ → ((♯‘𝑉) − 2) < ((♯‘𝑉) − 1))
27	23, 22	resubcld 11642	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑉) ∈ ℝ → ((♯‘𝑉) − 2) ∈ ℝ)
28		peano2rem 11527	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑉) ∈ ℝ → ((♯‘𝑉) − 1) ∈ ℝ)
29	27, 28	ltnled 11361	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑉) ∈ ℝ → (((♯‘𝑉) − 2) < ((♯‘𝑉) − 1) ↔ ¬ ((♯‘𝑉) − 1) ≤ ((♯‘𝑉) − 2)))
30	26, 29	mpbid 231	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑉) ∈ ℝ → ¬ ((♯‘𝑉) − 1) ≤ ((♯‘𝑉) − 2))
31	19, 30	syl 17	. . . . . . . . . . . 12 ⊢ ((♯‘𝑉) ∈ ℕ₀ → ¬ ((♯‘𝑉) − 1) ≤ ((♯‘𝑉) − 2))
32	17, 18, 31	3syl 18	. . . . . . . . . . 11 ⊢ (𝐺 ∈ FinUSGraph → ¬ ((♯‘𝑉) − 1) ≤ ((♯‘𝑉) − 2))
33	32	pm2.21d 121	. . . . . . . . . 10 ⊢ (𝐺 ∈ FinUSGraph → (((♯‘𝑉) − 1) ≤ ((♯‘𝑉) − 2) → 𝑀 ∈ (𝐺 NeighbVtx 𝑈)))
34	33	adantr 482	. . . . . . . . 9 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉) → (((♯‘𝑉) − 1) ≤ ((♯‘𝑉) − 2) → 𝑀 ∈ (𝐺 NeighbVtx 𝑈)))
35	34	ad3antlr 730	. . . . . . . 8 ⊢ ((((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉)) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈)) ∧ (♯‘(𝐺 NeighbVtx 𝑈)) = ((♯‘𝑉) − 1)) → (((♯‘𝑉) − 1) ≤ ((♯‘𝑉) − 2) → 𝑀 ∈ (𝐺 NeighbVtx 𝑈)))
36	16, 35	sylbid 239	. . . . . . 7 ⊢ ((((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉)) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈)) ∧ (♯‘(𝐺 NeighbVtx 𝑈)) = ((♯‘𝑉) − 1)) → ((♯‘(𝐺 NeighbVtx 𝑈)) ≤ ((♯‘𝑉) − 2) → 𝑀 ∈ (𝐺 NeighbVtx 𝑈)))
37	36	ex 414	. . . . . 6 ⊢ (((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉)) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈)) → ((♯‘(𝐺 NeighbVtx 𝑈)) = ((♯‘𝑉) − 1) → ((♯‘(𝐺 NeighbVtx 𝑈)) ≤ ((♯‘𝑉) − 2) → 𝑀 ∈ (𝐺 NeighbVtx 𝑈))))
38	14, 37	mpid 44	. . . . 5 ⊢ (((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉)) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈)) → ((♯‘(𝐺 NeighbVtx 𝑈)) = ((♯‘𝑉) − 1) → 𝑀 ∈ (𝐺 NeighbVtx 𝑈)))
39	38	ex 414	. . . 4 ⊢ ((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉)) → ((𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈) → ((♯‘(𝐺 NeighbVtx 𝑈)) = ((♯‘𝑉) − 1) → 𝑀 ∈ (𝐺 NeighbVtx 𝑈))))
40	39	com23 86	. . 3 ⊢ ((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉)) → ((♯‘(𝐺 NeighbVtx 𝑈)) = ((♯‘𝑉) − 1) → ((𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈) → 𝑀 ∈ (𝐺 NeighbVtx 𝑈))))
41	40	ex 414	. 2 ⊢ (¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) → ((𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉) → ((♯‘(𝐺 NeighbVtx 𝑈)) = ((♯‘𝑉) − 1) → ((𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈) → 𝑀 ∈ (𝐺 NeighbVtx 𝑈)))))
42	2, 41	pm2.61i 182	1 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉) → ((♯‘(𝐺 NeighbVtx 𝑈)) = ((♯‘𝑉) − 1) → ((𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈) → 𝑀 ∈ (𝐺 NeighbVtx 𝑈))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∉ wnel 3047 class class class wbr 5149 ‘cfv 6544 (class class class)co 7409 Fincfn 8939 ℝcr 11109 1c1 11111 < clt 11248 ≤ cle 11249 − cmin 11444 2c2 12267 ℕ₀cn0 12472 ♯chash 14290 Vtxcvtx 28256 FinUSGraphcfusgr 28573 NeighbVtx cnbgr 28589

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-oadd 8470 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-dju 9896 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-2 12275 df-n0 12473 df-xnn0 12545 df-z 12559 df-uz 12823 df-fz 13485 df-hash 14291 df-fusgr 28574 df-nbgr 28590

This theorem is referenced by: nbusgrvtxm1uvtx 28662

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