Theorem frgrwopreglem4a 30246

Description: In a friendship graph any two vertices with different degrees are connected. Alternate version of frgrwopreglem4 30251 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 6861	. . . . . 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 2947	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((𝐷‘𝑋) ≠ (𝐷‘𝑌) → 𝑋 ≠ 𝑌))
4	3	imp 406	. . 3 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ≠ (𝐷‘𝑌)) → 𝑋 ≠ 𝑌)
5	4	3adant1 1130	. 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 30245	. . . . . 6 ⊢ (𝐺 ∈ FriendGraph → ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷‘𝑥) = (𝐷‘𝑦)))
9		oveq2 7398	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (𝐺 NeighbVtx 𝑥) = (𝐺 NeighbVtx 𝑋))
10		neleq2 3037	. . . . . . . . . . 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 6870	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → ((𝐷‘𝑥) = (𝐷‘𝑦) ↔ (𝐷‘𝑋) = (𝐷‘𝑦)))
13	11, 12	imbi12d 344	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → ((𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷‘𝑥) = (𝐷‘𝑦)) ↔ (𝑦 ∉ (𝐺 NeighbVtx 𝑋) → (𝐷‘𝑋) = (𝐷‘𝑦))))
14		neleq1 3036	. . . . . . . . . 10 ⊢ (𝑦 = 𝑌 → (𝑦 ∉ (𝐺 NeighbVtx 𝑋) ↔ 𝑌 ∉ (𝐺 NeighbVtx 𝑋)))
15		fveq2 6861	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑌 → (𝐷‘𝑦) = (𝐷‘𝑌))
16	15	eqeq2d 2741	. . . . . . . . . 10 ⊢ (𝑦 = 𝑌 → ((𝐷‘𝑋) = (𝐷‘𝑦) ↔ (𝐷‘𝑋) = (𝐷‘𝑌)))
17	14, 16	imbi12d 344	. . . . . . . . 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 4602	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋})
20	19	difeq2d 4092	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (𝑉 ∖ {𝑥}) = (𝑉 ∖ {𝑋}))
21	20	adantl 481	. . . . . . . . 9 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ 𝑋 ≠ 𝑌) ∧ 𝑥 = 𝑋) → (𝑉 ∖ {𝑥}) = (𝑉 ∖ {𝑋}))
22		simpr 484	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → 𝑌 ∈ 𝑉)
23		necom 2979	. . . . . . . . . . . 12 ⊢ (𝑋 ≠ 𝑌 ↔ 𝑌 ≠ 𝑋)
24	23	biimpi 216	. . . . . . . . . . 11 ⊢ (𝑋 ≠ 𝑌 → 𝑌 ≠ 𝑋)
25	22, 24	anim12i 613	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ 𝑋 ≠ 𝑌) → (𝑌 ∈ 𝑉 ∧ 𝑌 ≠ 𝑋))
26		eldifsn 4753	. . . . . . . . . 10 ⊢ (𝑌 ∈ (𝑉 ∖ {𝑋}) ↔ (𝑌 ∈ 𝑉 ∧ 𝑌 ≠ 𝑋))
27	25, 26	sylibr 234	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ 𝑋 ≠ 𝑌) → 𝑌 ∈ (𝑉 ∖ {𝑋}))
28	13, 17, 18, 21, 27	rspc2vd 3913	. . . . . . . 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 3040	. . . . . . . . . . 11 ⊢ (¬ 𝑌 ∉ (𝐺 NeighbVtx 𝑋) ↔ 𝑌 ∈ (𝐺 NeighbVtx 𝑋))
30		nbgrsym 29297	. . . . . . . . . . . . . . . 16 ⊢ (𝑌 ∈ (𝐺 NeighbVtx 𝑋) ↔ 𝑋 ∈ (𝐺 NeighbVtx 𝑌))
31		frgrusgr 30197	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
32		frgrwopreglem4a.e	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐸 = (Edg‘𝐺)
33	32	nbusgreledg 29287	. . . . . . . . . . . . . . . . . 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 229	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ FriendGraph → (𝑋 ∈ (𝐺 NeighbVtx 𝑌) → {𝑋, 𝑌} ∈ 𝐸))
36	30, 35	biimtrid 242	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ FriendGraph → (𝑌 ∈ (𝐺 NeighbVtx 𝑋) → {𝑋, 𝑌} ∈ 𝐸))
37	36	imp 406	. . . . . . . . . . . . . 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 413	. . . . . . . . . . . 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 217	. . . . . . . . . 10 ⊢ (¬ 𝑌 ∉ (𝐺 NeighbVtx 𝑋) → (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ 𝑋 ≠ 𝑌) → (𝐺 ∈ FriendGraph → ((𝐷‘𝑋) ≠ (𝐷‘𝑌) → {𝑋, 𝑌} ∈ 𝐸))))
42		eqneqall 2937	. . . . . . . . . . 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 186	. . . . . . . . 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 415	. . . 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 1110	. 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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2926   ∉ wnel 3030  ∀wral 3045   ∖ cdif 3914  {csn 4592  {cpr 4594  ‘cfv 6514  (class class class)co 7390  Vtxcvtx 28930  Edgcedg 28981  USGraphcusgr 29083   NeighbVtx cnbgr 29266  VtxDegcvtxdg 29400   FriendGraph cfrgr 30194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-2o 8438  df-oadd 8441  df-er 8674  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-dju 9861  df-card 9899  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-nn 12194  df-2 12256  df-n0 12450  df-xnn0 12523  df-z 12537  df-uz 12801  df-xadd 13080  df-fz 13476  df-hash 14303  df-edg 28982  df-uhgr 28992  df-ushgr 28993  df-upgr 29016  df-umgr 29017  df-uspgr 29084  df-usgr 29085  df-nbgr 29267  df-vtxdg 29401  df-frgr 30195

This theorem is referenced by:  frgrwopreglem5a  30247  frgrwopreglem4  30251