Theorem frgrwopreglem4a 30342

Description: In a friendship graph any two vertices with different degrees are connected. Alternate version of frgrwopreglem4 30347 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 6920	. . . . . 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 2967	. . . 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 30341	. . . . . 6 ⊢ (𝐺 ∈ FriendGraph → ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷‘𝑥) = (𝐷‘𝑦)))
9		oveq2 7456	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (𝐺 NeighbVtx 𝑥) = (𝐺 NeighbVtx 𝑋))
10		neleq2 3059	. . . . . . . . . . 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 6929	. . . . . . . . . 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 3058	. . . . . . . . . 10 ⊢ (𝑦 = 𝑌 → (𝑦 ∉ (𝐺 NeighbVtx 𝑋) ↔ 𝑌 ∉ (𝐺 NeighbVtx 𝑋)))
15		fveq2 6920	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑌 → (𝐷‘𝑦) = (𝐷‘𝑌))
16	15	eqeq2d 2751	. . . . . . . . . 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 4658	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋})
20	19	difeq2d 4149	. . . . . . . . . 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 3000	. . . . . . . . . . . 12 ⊢ (𝑋 ≠ 𝑌 ↔ 𝑌 ≠ 𝑋)
24	23	biimpi 216	. . . . . . . . . . 11 ⊢ (𝑋 ≠ 𝑌 → 𝑌 ≠ 𝑋)
25	22, 24	anim12i 612	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ 𝑋 ≠ 𝑌) → (𝑌 ∈ 𝑉 ∧ 𝑌 ≠ 𝑋))
26		eldifsn 4811	. . . . . . . . . 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 3972	. . . . . . . 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 3062	. . . . . . . . . . 11 ⊢ (¬ 𝑌 ∉ (𝐺 NeighbVtx 𝑋) ↔ 𝑌 ∈ (𝐺 NeighbVtx 𝑋))
30		nbgrsym 29398	. . . . . . . . . . . . . . . 16 ⊢ (𝑌 ∈ (𝐺 NeighbVtx 𝑋) ↔ 𝑋 ∈ (𝐺 NeighbVtx 𝑌))
31		frgrusgr 30293	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
32		frgrwopreglem4a.e	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐸 = (Edg‘𝐺)
33	32	nbusgreledg 29388	. . . . . . . . . . . . . . . . . 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 2957	. . . . . . . . . . 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 1111	. 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 1087   = wceq 1537   ∈ wcel 2108   ≠ wne 2946   ∉ wnel 3052  ∀wral 3067   ∖ cdif 3973  {csn 4648  {cpr 4650  ‘cfv 6573  (class class class)co 7448  Vtxcvtx 29031  Edgcedg 29082  USGraphcusgr 29184   NeighbVtx cnbgr 29367  VtxDegcvtxdg 29501   FriendGraph cfrgr 30290

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-oadd 8526  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-dju 9970  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-n0 12554  df-xnn0 12626  df-z 12640  df-uz 12904  df-xadd 13176  df-fz 13568  df-hash 14380  df-edg 29083  df-uhgr 29093  df-ushgr 29094  df-upgr 29117  df-umgr 29118  df-uspgr 29185  df-usgr 29186  df-nbgr 29368  df-vtxdg 29502  df-frgr 30291

This theorem is referenced by:  frgrwopreglem5a  30343  frgrwopreglem4  30347