Theorem frgrwopreglem4a 30602

Description: In a friendship graph any two vertices with different degrees are connected. Alternate version of frgrwopreglem4 30607 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 6882	. . . . . 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 2985	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((𝐷‘𝑋) ≠ (𝐷‘𝑌) → 𝑋 ≠ 𝑌))
4	3	imp 411	. . 3 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ≠ (𝐷‘𝑌)) → 𝑋 ≠ 𝑌)
5	4	3adant1 1146	. 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 30601	. . . . . 6 ⊢ (𝐺 ∈ FriendGraph → ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷‘𝑥) = (𝐷‘𝑦)))
9		oveq2 7419	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (𝐺 NeighbVtx 𝑥) = (𝐺 NeighbVtx 𝑋))
10		neleq2 3077	. . . . . . . . . . 11 ⊢ ((𝐺 NeighbVtx 𝑥) = (𝐺 NeighbVtx 𝑋) → (𝑦 ∉ (𝐺 NeighbVtx 𝑥) ↔ 𝑦 ∉ (𝐺 NeighbVtx 𝑋)))
11	9, 10	syl 18	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (𝑦 ∉ (𝐺 NeighbVtx 𝑥) ↔ 𝑦 ∉ (𝐺 NeighbVtx 𝑋)))
12		fveqeq2 6891	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → ((𝐷‘𝑥) = (𝐷‘𝑦) ↔ (𝐷‘𝑋) = (𝐷‘𝑦)))
13	11, 12	imbi12d 347	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → ((𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷‘𝑥) = (𝐷‘𝑦)) ↔ (𝑦 ∉ (𝐺 NeighbVtx 𝑋) → (𝐷‘𝑋) = (𝐷‘𝑦))))
14		neleq1 3076	. . . . . . . . . 10 ⊢ (𝑦 = 𝑌 → (𝑦 ∉ (𝐺 NeighbVtx 𝑋) ↔ 𝑌 ∉ (𝐺 NeighbVtx 𝑋)))
15		fveq2 6882	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑌 → (𝐷‘𝑦) = (𝐷‘𝑌))
16	15	eqeq2d 2780	. . . . . . . . . 10 ⊢ (𝑦 = 𝑌 → ((𝐷‘𝑋) = (𝐷‘𝑦) ↔ (𝐷‘𝑋) = (𝐷‘𝑌)))
17	14, 16	imbi12d 347	. . . . . . . . 9 ⊢ (𝑦 = 𝑌 → ((𝑦 ∉ (𝐺 NeighbVtx 𝑋) → (𝐷‘𝑋) = (𝐷‘𝑦)) ↔ (𝑌 ∉ (𝐺 NeighbVtx 𝑋) → (𝐷‘𝑋) = (𝐷‘𝑌))))
18		simpll 778	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ 𝑋 ≠ 𝑌) → 𝑋 ∈ 𝑉)
19		sneq 4602	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋})
20	19	difeq2d 4087	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (𝑉 ∖ {𝑥}) = (𝑉 ∖ {𝑋}))
21	20	adantl 486	. . . . . . . . 9 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ 𝑋 ≠ 𝑌) ∧ 𝑥 = 𝑋) → (𝑉 ∖ {𝑥}) = (𝑉 ∖ {𝑋}))
22		simpr 489	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → 𝑌 ∈ 𝑉)
23		necom 3017	. . . . . . . . . . . 12 ⊢ (𝑋 ≠ 𝑌 ↔ 𝑌 ≠ 𝑋)
24	23	biimpi 219	. . . . . . . . . . 11 ⊢ (𝑋 ≠ 𝑌 → 𝑌 ≠ 𝑋)
25	22, 24	anim12i 624	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ 𝑋 ≠ 𝑌) → (𝑌 ∈ 𝑉 ∧ 𝑌 ≠ 𝑋))
26		eldifsn 4756	. . . . . . . . . 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 3907	. . . . . . . 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 3080	. . . . . . . . . . 11 ⊢ (¬ 𝑌 ∉ (𝐺 NeighbVtx 𝑋) ↔ 𝑌 ∈ (𝐺 NeighbVtx 𝑋))
30		nbgrsym 29654	. . . . . . . . . . . . . . . 16 ⊢ (𝑌 ∈ (𝐺 NeighbVtx 𝑋) ↔ 𝑋 ∈ (𝐺 NeighbVtx 𝑌))
31		frgrusgr 30553	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
32		frgrwopreglem4a.e	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐸 = (Edg‘𝐺)
33	32	nbusgreledg 29644	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 ∈ USGraph → (𝑋 ∈ (𝐺 NeighbVtx 𝑌) ↔ {𝑋, 𝑌} ∈ 𝐸))
34	31, 33	syl 18	. . . . . . . . . . . . . . . . 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	biimtrid 245	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ FriendGraph → (𝑌 ∈ (𝐺 NeighbVtx 𝑋) → {𝑋, 𝑌} ∈ 𝐸))
37	36	imp 411	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑌 ∈ (𝐺 NeighbVtx 𝑋)) → {𝑋, 𝑌} ∈ 𝐸)
38	37	a1d 26	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑌 ∈ (𝐺 NeighbVtx 𝑋)) → ((𝐷‘𝑋) ≠ (𝐷‘𝑌) → {𝑋, 𝑌} ∈ 𝐸))
39	38	expcom 418	. . . . . . . . . . . 12 ⊢ (𝑌 ∈ (𝐺 NeighbVtx 𝑋) → (𝐺 ∈ FriendGraph → ((𝐷‘𝑋) ≠ (𝐷‘𝑌) → {𝑋, 𝑌} ∈ 𝐸)))
40	39	a1d 26	. . . . . . . . . . 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 2975	. . . . . . . . . . 11 ⊢ ((𝐷‘𝑋) = (𝐷‘𝑌) → ((𝐷‘𝑋) ≠ (𝐷‘𝑌) → {𝑋, 𝑌} ∈ 𝐸))
43	42	2a1d 27	. . . . . . . . . 10 ⊢ ((𝐷‘𝑋) = (𝐷‘𝑌) → (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ 𝑋 ≠ 𝑌) → (𝐺 ∈ FriendGraph → ((𝐷‘𝑋) ≠ (𝐷‘𝑌) → {𝑋, 𝑌} ∈ 𝐸))))
44	41, 43	ja 188	. . . . . . . . 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 33	. . . . . . . 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 48	. . . . . . 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 90	. . . . . 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 39	. . . . 5 ⊢ (𝐺 ∈ FriendGraph → (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ 𝑋 ≠ 𝑌) → ((𝐷‘𝑋) ≠ (𝐷‘𝑌) → {𝑋, 𝑌} ∈ 𝐸)))
49	48	expd 420	. . . 4 ⊢ (𝐺 ∈ FriendGraph → ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 ≠ 𝑌 → ((𝐷‘𝑋) ≠ (𝐷‘𝑌) → {𝑋, 𝑌} ∈ 𝐸))))
50	49	com34 92	. . 3 ⊢ (𝐺 ∈ FriendGraph → ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((𝐷‘𝑋) ≠ (𝐷‘𝑌) → (𝑋 ≠ 𝑌 → {𝑋, 𝑌} ∈ 𝐸))))
51	50	3imp 1126	. 2 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ≠ (𝐷‘𝑌)) → (𝑋 ≠ 𝑌 → {𝑋, 𝑌} ∈ 𝐸))
52	5, 51	mpd 16	1 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ≠ (𝐷‘𝑌)) → {𝑋, 𝑌} ∈ 𝐸)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149   ≠ wne 2964   ∉ wnel 3070  ∀wral 3085   ∖ cdif 3908  {csn 4592  {cpr 4594  ‘cfv 6537  (class class class)co 7411  Vtxcvtx 29287  Edgcedg 29338  USGraphcusgr 29440   NeighbVtx cnbgr 29623  VtxDegcvtxdg 29756   FriendGraph cfrgr 30550

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11156  ax-resscn 11157  ax-1cn 11158  ax-icn 11159  ax-addcl 11160  ax-addrcl 11161  ax-mulcl 11162  ax-mulrcl 11163  ax-mulcom 11164  ax-addass 11165  ax-mulass 11166  ax-distr 11167  ax-i2m1 11168  ax-1ne0 11169  ax-1rid 11170  ax-rnegex 11171  ax-rrecex 11172  ax-cnre 11173  ax-pre-lttri 11174  ax-pre-lttrn 11175  ax-pre-ltadd 11176  ax-pre-mulgt0 11177

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3375  df-reu 3376  df-rab 3423  df-v 3463  df-sbc 3752  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4491  df-pw 4567  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8453  df-2o 8454  df-oadd 8457  df-er 8694  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-dju 9887  df-card 9925  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-sub 11443  df-neg 11444  df-nn 12234  df-2 12303  df-n0 12505  df-xnn0 12578  df-z 12592  df-uz 12863  df-xadd 13138  df-fz 13536  df-hash 14367  df-edg 29339  df-uhgr 29349  df-ushgr 29350  df-upgr 29373  df-umgr 29374  df-uspgr 29441  df-usgr 29442  df-nbgr 29624  df-vtxdg 29757  df-frgr 30551

This theorem is referenced by:  frgrwopreglem5a  30603  frgrwopreglem4  30607