Theorem frgrwopreglem4a 28194

Description: In a friendship graph any two vertices with different degrees are connected. Alternate version of frgrwopreglem4 28199 without a fixed degree and without using the sets 𝐴 and 𝐵. (Contributed by Alexander van der Vekens, 30-Dec-2017.) (Revised by AV, 4-Feb-2022.)

Hypotheses

Ref	Expression
frgrncvvdeq.v	⊢ 𝑉 = (Vtx‘𝐺)
frgrncvvdeq.d	⊢ 𝐷 = (VtxDeg‘𝐺)
frgrwopreglem4a.e	⊢ 𝐸 = (Edg‘𝐺)

Assertion

Ref	Expression
frgrwopreglem4a	⊢ ((𝐺 ∈ FriendGraph ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ≠ (𝐷‘𝑌)) → {𝑋, 𝑌} ∈ 𝐸)

Proof of Theorem frgrwopreglem4a

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6658	. . . . . 6 ⊢ (𝑋 = 𝑌 → (𝐷‘𝑋) = (𝐷‘𝑌))
2	1	a1i 11	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 = 𝑌 → (𝐷‘𝑋) = (𝐷‘𝑌)))
3	2	necon3d 2972	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((𝐷‘𝑋) ≠ (𝐷‘𝑌) → 𝑋 ≠ 𝑌))
4	3	imp 410	. . 3 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ≠ (𝐷‘𝑌)) → 𝑋 ≠ 𝑌)
5	4	3adant1 1127	. 2 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ≠ (𝐷‘𝑌)) → 𝑋 ≠ 𝑌)
6		frgrncvvdeq.v	. . . . . . 7 ⊢ 𝑉 = (Vtx‘𝐺)
7		frgrncvvdeq.d	. . . . . . 7 ⊢ 𝐷 = (VtxDeg‘𝐺)
8	6, 7	frgrncvvdeq 28193	. . . . . 6 ⊢ (𝐺 ∈ FriendGraph → ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷‘𝑥) = (𝐷‘𝑦)))
9		oveq2 7158	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (𝐺 NeighbVtx 𝑥) = (𝐺 NeighbVtx 𝑋))
10		neleq2 3061	. . . . . . . . . . 11 ⊢ ((𝐺 NeighbVtx 𝑥) = (𝐺 NeighbVtx 𝑋) → (𝑦 ∉ (𝐺 NeighbVtx 𝑥) ↔ 𝑦 ∉ (𝐺 NeighbVtx 𝑋)))
11	9, 10	syl 17	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (𝑦 ∉ (𝐺 NeighbVtx 𝑥) ↔ 𝑦 ∉ (𝐺 NeighbVtx 𝑋)))
12		fveqeq2 6667	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → ((𝐷‘𝑥) = (𝐷‘𝑦) ↔ (𝐷‘𝑋) = (𝐷‘𝑦)))
13	11, 12	imbi12d 348	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → ((𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷‘𝑥) = (𝐷‘𝑦)) ↔ (𝑦 ∉ (𝐺 NeighbVtx 𝑋) → (𝐷‘𝑋) = (𝐷‘𝑦))))
14		neleq1 3060	. . . . . . . . . 10 ⊢ (𝑦 = 𝑌 → (𝑦 ∉ (𝐺 NeighbVtx 𝑋) ↔ 𝑌 ∉ (𝐺 NeighbVtx 𝑋)))
15		fveq2 6658	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑌 → (𝐷‘𝑦) = (𝐷‘𝑌))
16	15	eqeq2d 2769	. . . . . . . . . 10 ⊢ (𝑦 = 𝑌 → ((𝐷‘𝑋) = (𝐷‘𝑦) ↔ (𝐷‘𝑋) = (𝐷‘𝑌)))
17	14, 16	imbi12d 348	. . . . . . . . 9 ⊢ (𝑦 = 𝑌 → ((𝑦 ∉ (𝐺 NeighbVtx 𝑋) → (𝐷‘𝑋) = (𝐷‘𝑦)) ↔ (𝑌 ∉ (𝐺 NeighbVtx 𝑋) → (𝐷‘𝑋) = (𝐷‘𝑌))))
18		simpll 766	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ 𝑋 ≠ 𝑌) → 𝑋 ∈ 𝑉)
19		sneq 4532	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋})
20	19	difeq2d 4028	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (𝑉 ∖ {𝑥}) = (𝑉 ∖ {𝑋}))
21	20	adantl 485	. . . . . . . . 9 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ 𝑋 ≠ 𝑌) ∧ 𝑥 = 𝑋) → (𝑉 ∖ {𝑥}) = (𝑉 ∖ {𝑋}))
22		simpr 488	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → 𝑌 ∈ 𝑉)
23		necom 3004	. . . . . . . . . . . 12 ⊢ (𝑋 ≠ 𝑌 ↔ 𝑌 ≠ 𝑋)
24	23	biimpi 219	. . . . . . . . . . 11 ⊢ (𝑋 ≠ 𝑌 → 𝑌 ≠ 𝑋)
25	22, 24	anim12i 615	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ 𝑋 ≠ 𝑌) → (𝑌 ∈ 𝑉 ∧ 𝑌 ≠ 𝑋))
26		eldifsn 4677	. . . . . . . . . 10 ⊢ (𝑌 ∈ (𝑉 ∖ {𝑋}) ↔ (𝑌 ∈ 𝑉 ∧ 𝑌 ≠ 𝑋))
27	25, 26	sylibr 237	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ 𝑋 ≠ 𝑌) → 𝑌 ∈ (𝑉 ∖ {𝑋}))
28	13, 17, 18, 21, 27	rspc2vd 3854	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ 𝑋 ≠ 𝑌) → (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷‘𝑥) = (𝐷‘𝑦)) → (𝑌 ∉ (𝐺 NeighbVtx 𝑋) → (𝐷‘𝑋) = (𝐷‘𝑌))))
29		nnel 3064	. . . . . . . . . . 11 ⊢ (¬ 𝑌 ∉ (𝐺 NeighbVtx 𝑋) ↔ 𝑌 ∈ (𝐺 NeighbVtx 𝑋))
30		nbgrsym 27252	. . . . . . . . . . . . . . . 16 ⊢ (𝑌 ∈ (𝐺 NeighbVtx 𝑋) ↔ 𝑋 ∈ (𝐺 NeighbVtx 𝑌))
31		frgrusgr 28145	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
32		frgrwopreglem4a.e	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐸 = (Edg‘𝐺)
33	32	nbusgreledg 27242	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 ∈ USGraph → (𝑋 ∈ (𝐺 NeighbVtx 𝑌) ↔ {𝑋, 𝑌} ∈ 𝐸))
34	31, 33	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺 ∈ FriendGraph → (𝑋 ∈ (𝐺 NeighbVtx 𝑌) ↔ {𝑋, 𝑌} ∈ 𝐸))
35	34	biimpd 232	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ FriendGraph → (𝑋 ∈ (𝐺 NeighbVtx 𝑌) → {𝑋, 𝑌} ∈ 𝐸))
36	30, 35	syl5bi 245	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ FriendGraph → (𝑌 ∈ (𝐺 NeighbVtx 𝑋) → {𝑋, 𝑌} ∈ 𝐸))
37	36	imp 410	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑌 ∈ (𝐺 NeighbVtx 𝑋)) → {𝑋, 𝑌} ∈ 𝐸)
38	37	a1d 25	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑌 ∈ (𝐺 NeighbVtx 𝑋)) → ((𝐷‘𝑋) ≠ (𝐷‘𝑌) → {𝑋, 𝑌} ∈ 𝐸))
39	38	expcom 417	. . . . . . . . . . . 12 ⊢ (𝑌 ∈ (𝐺 NeighbVtx 𝑋) → (𝐺 ∈ FriendGraph → ((𝐷‘𝑋) ≠ (𝐷‘𝑌) → {𝑋, 𝑌} ∈ 𝐸)))
40	39	a1d 25	. . . . . . . . . . 11 ⊢ (𝑌 ∈ (𝐺 NeighbVtx 𝑋) → (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ 𝑋 ≠ 𝑌) → (𝐺 ∈ FriendGraph → ((𝐷‘𝑋) ≠ (𝐷‘𝑌) → {𝑋, 𝑌} ∈ 𝐸))))
41	29, 40	sylbi 220	. . . . . . . . . 10 ⊢ (¬ 𝑌 ∉ (𝐺 NeighbVtx 𝑋) → (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ 𝑋 ≠ 𝑌) → (𝐺 ∈ FriendGraph → ((𝐷‘𝑋) ≠ (𝐷‘𝑌) → {𝑋, 𝑌} ∈ 𝐸))))
42		eqneqall 2962	. . . . . . . . . . 11 ⊢ ((𝐷‘𝑋) = (𝐷‘𝑌) → ((𝐷‘𝑋) ≠ (𝐷‘𝑌) → {𝑋, 𝑌} ∈ 𝐸))
43	42	2a1d 26	. . . . . . . . . 10 ⊢ ((𝐷‘𝑋) = (𝐷‘𝑌) → (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ 𝑋 ≠ 𝑌) → (𝐺 ∈ FriendGraph → ((𝐷‘𝑋) ≠ (𝐷‘𝑌) → {𝑋, 𝑌} ∈ 𝐸))))
44	41, 43	ja 189	. . . . . . . . 9 ⊢ ((𝑌 ∉ (𝐺 NeighbVtx 𝑋) → (𝐷‘𝑋) = (𝐷‘𝑌)) → (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ 𝑋 ≠ 𝑌) → (𝐺 ∈ FriendGraph → ((𝐷‘𝑋) ≠ (𝐷‘𝑌) → {𝑋, 𝑌} ∈ 𝐸))))
45	44	com12 32	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ 𝑋 ≠ 𝑌) → ((𝑌 ∉ (𝐺 NeighbVtx 𝑋) → (𝐷‘𝑋) = (𝐷‘𝑌)) → (𝐺 ∈ FriendGraph → ((𝐷‘𝑋) ≠ (𝐷‘𝑌) → {𝑋, 𝑌} ∈ 𝐸))))
46	28, 45	syld 47	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ 𝑋 ≠ 𝑌) → (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷‘𝑥) = (𝐷‘𝑦)) → (𝐺 ∈ FriendGraph → ((𝐷‘𝑋) ≠ (𝐷‘𝑌) → {𝑋, 𝑌} ∈ 𝐸))))
47	46	com3l 89	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷‘𝑥) = (𝐷‘𝑦)) → (𝐺 ∈ FriendGraph → (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ 𝑋 ≠ 𝑌) → ((𝐷‘𝑋) ≠ (𝐷‘𝑌) → {𝑋, 𝑌} ∈ 𝐸))))
48	8, 47	mpcom 38	. . . . 5 ⊢ (𝐺 ∈ FriendGraph → (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ 𝑋 ≠ 𝑌) → ((𝐷‘𝑋) ≠ (𝐷‘𝑌) → {𝑋, 𝑌} ∈ 𝐸)))
49	48	expd 419	. . . 4 ⊢ (𝐺 ∈ FriendGraph → ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 ≠ 𝑌 → ((𝐷‘𝑋) ≠ (𝐷‘𝑌) → {𝑋, 𝑌} ∈ 𝐸))))
50	49	com34 91	. . 3 ⊢ (𝐺 ∈ FriendGraph → ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((𝐷‘𝑋) ≠ (𝐷‘𝑌) → (𝑋 ≠ 𝑌 → {𝑋, 𝑌} ∈ 𝐸))))
51	50	3imp 1108	. 2 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ≠ (𝐷‘𝑌)) → (𝑋 ≠ 𝑌 → {𝑋, 𝑌} ∈ 𝐸))
52	5, 51	mpd 15	1 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ≠ (𝐷‘𝑌)) → {𝑋, 𝑌} ∈ 𝐸)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2951   ∉ wnel 3055  ∀wral 3070   ∖ cdif 3855  {csn 4522  {cpr 4524  ‘cfv 6335  (class class class)co 7150  Vtxcvtx 26888  Edgcedg 26939  USGraphcusgr 27041   NeighbVtx cnbgr 27221  VtxDegcvtxdg 27354   FriendGraph cfrgr 28142

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459  ax-cnex 10631  ax-resscn 10632  ax-1cn 10633  ax-icn 10634  ax-addcl 10635  ax-addrcl 10636  ax-mulcl 10637  ax-mulrcl 10638  ax-mulcom 10639  ax-addass 10640  ax-mulass 10641  ax-distr 10642  ax-i2m1 10643  ax-1ne0 10644  ax-1rid 10645  ax-rnegex 10646  ax-rrecex 10647  ax-cnre 10648  ax-pre-lttri 10649  ax-pre-lttrn 10650  ax-pre-ltadd 10651  ax-pre-mulgt0 10652

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-int 4839  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7580  df-1st 7693  df-2nd 7694  df-wrecs 7957  df-recs 8018  df-rdg 8056  df-1o 8112  df-2o 8113  df-oadd 8116  df-er 8299  df-en 8528  df-dom 8529  df-sdom 8530  df-fin 8531  df-dju 9363  df-card 9401  df-pnf 10715  df-mnf 10716  df-xr 10717  df-ltxr 10718  df-le 10719  df-sub 10910  df-neg 10911  df-nn 11675  df-2 11737  df-n0 11935  df-xnn0 12007  df-z 12021  df-uz 12283  df-xadd 12549  df-fz 12940  df-hash 13741  df-edg 26940  df-uhgr 26950  df-ushgr 26951  df-upgr 26974  df-umgr 26975  df-uspgr 27042  df-usgr 27043  df-nbgr 27222  df-vtxdg 27355  df-frgr 28143

This theorem is referenced by:  frgrwopreglem5a  28195  frgrwopreglem4  28199