Theorem frgrwopreglem4a 30338

Description: In a friendship graph any two vertices with different degrees are connected. Alternate version of frgrwopreglem4 30343 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 6906	. . . . . 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 2958	. . . 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 1129	. 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 30337	. . . . . 6 ⊢ (𝐺 ∈ FriendGraph → ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷‘𝑥) = (𝐷‘𝑦)))
9		oveq2 7438	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (𝐺 NeighbVtx 𝑥) = (𝐺 NeighbVtx 𝑋))
10		neleq2 3050	. . . . . . . . . . 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 6915	. . . . . . . . . 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 3049	. . . . . . . . . 10 ⊢ (𝑦 = 𝑌 → (𝑦 ∉ (𝐺 NeighbVtx 𝑋) ↔ 𝑌 ∉ (𝐺 NeighbVtx 𝑋)))
15		fveq2 6906	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑌 → (𝐷‘𝑦) = (𝐷‘𝑌))
16	15	eqeq2d 2745	. . . . . . . . . 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 767	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ 𝑋 ≠ 𝑌) → 𝑋 ∈ 𝑉)
19		sneq 4640	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋})
20	19	difeq2d 4135	. . . . . . . . . 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 2991	. . . . . . . . . . . 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 4790	. . . . . . . . . 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 3958	. . . . . . . 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 3053	. . . . . . . . . . 11 ⊢ (¬ 𝑌 ∉ (𝐺 NeighbVtx 𝑋) ↔ 𝑌 ∈ (𝐺 NeighbVtx 𝑋))
30		nbgrsym 29394	. . . . . . . . . . . . . . . 16 ⊢ (𝑌 ∈ (𝐺 NeighbVtx 𝑋) ↔ 𝑋 ∈ (𝐺 NeighbVtx 𝑌))
31		frgrusgr 30289	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
32		frgrwopreglem4a.e	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐸 = (Edg‘𝐺)
33	32	nbusgreledg 29384	. . . . . . . . . . . . . . . . . 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 2948	. . . . . . . . . . 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 1536   ∈ wcel 2105   ≠ wne 2937   ∉ wnel 3043  ∀wral 3058   ∖ cdif 3959  {csn 4630  {cpr 4632  ‘cfv 6562  (class class class)co 7430  Vtxcvtx 29027  Edgcedg 29078  USGraphcusgr 29180   NeighbVtx cnbgr 29363  VtxDegcvtxdg 29497   FriendGraph cfrgr 30286

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-oadd 8508  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-dju 9938  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-n0 12524  df-xnn0 12597  df-z 12611  df-uz 12876  df-xadd 13152  df-fz 13544  df-hash 14366  df-edg 29079  df-uhgr 29089  df-ushgr 29090  df-upgr 29113  df-umgr 29114  df-uspgr 29181  df-usgr 29182  df-nbgr 29364  df-vtxdg 29498  df-frgr 30287

This theorem is referenced by:  frgrwopreglem5a  30339  frgrwopreglem4  30343