Theorem frgrwopreglem4a 27174

Description: In a friendship graph any two vertices with different degrees are connected. Alternate version of frgrwopreglem4 27179 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 e. FriendGraph /\ ( X e. V /\ Y e. V ) /\ ( D ` X ) =/= ( D ` Y ) ) -> { X , Y } e. E )

Proof of Theorem frgrwopreglem4a

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

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . . . 6 |- ( X = Y -> ( D ` X ) = ( D ` Y ) )
2	1	a1i 11	. . . . 5 |- ( ( X e. V /\ Y e. V ) -> ( X = Y -> ( D ` X ) = ( D ` Y ) ) )
3	2	necon3d 2815	. . . 4 |- ( ( X e. V /\ Y e. V ) -> ( ( D ` X ) =/= ( D ` Y ) -> X =/= Y ) )
4	3	imp 445	. . 3 |- ( ( ( X e. V /\ Y e. V ) /\ ( D ` X ) =/= ( D ` Y ) ) -> X =/= Y )
5	4	3adant1 1079	. 2 |- ( ( G e. FriendGraph /\ ( X e. V /\ Y e. 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 27173	. . . . . 6 |- ( G e. FriendGraph -> A. x e. V A. y e. ( V \ { x } ) ( y e/ ( G NeighbVtx x ) -> ( D ` x ) = ( D ` y ) ) )
9		oveq2 6658	. . . . . . . . . . 11 |- ( x = X -> ( G NeighbVtx x ) = ( G NeighbVtx X ) )
10		neleq2 2903	. . . . . . . . . . 11 |- ( ( G NeighbVtx x ) = ( G NeighbVtx X ) -> ( y e/ ( G NeighbVtx x ) <-> y e/ ( G NeighbVtx X ) ) )
11	9, 10	syl 17	. . . . . . . . . 10 |- ( x = X -> ( y e/ ( G NeighbVtx x ) <-> y e/ ( G NeighbVtx X ) ) )
12		fveq2 6191	. . . . . . . . . . 11 |- ( x = X -> ( D ` x ) = ( D ` X ) )
13	12	eqeq1d 2624	. . . . . . . . . 10 |- ( x = X -> ( ( D ` x ) = ( D ` y ) <-> ( D ` X ) = ( D ` y ) ) )
14	11, 13	imbi12d 334	. . . . . . . . 9 |- ( x = X -> ( ( y e/ ( G NeighbVtx x ) -> ( D ` x ) = ( D ` y ) ) <-> ( y e/ ( G NeighbVtx X ) -> ( D ` X ) = ( D ` y ) ) ) )
15		neleq1 2902	. . . . . . . . . 10 |- ( y = Y -> ( y e/ ( G NeighbVtx X ) <-> Y e/ ( G NeighbVtx X ) ) )
16		fveq2 6191	. . . . . . . . . . 11 |- ( y = Y -> ( D ` y ) = ( D ` Y ) )
17	16	eqeq2d 2632	. . . . . . . . . 10 |- ( y = Y -> ( ( D ` X ) = ( D ` y ) <-> ( D ` X ) = ( D ` Y ) ) )
18	15, 17	imbi12d 334	. . . . . . . . 9 |- ( y = Y -> ( ( y e/ ( G NeighbVtx X ) -> ( D ` X ) = ( D ` y ) ) <-> ( Y e/ ( G NeighbVtx X ) -> ( D ` X ) = ( D ` Y ) ) ) )
19		simpll 790	. . . . . . . . 9 |- ( ( ( X e. V /\ Y e. V ) /\ X =/= Y ) -> X e. V )
20		sneq 4187	. . . . . . . . . . 11 |- ( x = X -> { x } = { X } )
21	20	difeq2d 3728	. . . . . . . . . 10 |- ( x = X -> ( V \ { x } ) = ( V \ { X } ) )
22	21	adantl 482	. . . . . . . . 9 |- ( ( ( ( X e. V /\ Y e. V ) /\ X =/= Y ) /\ x = X ) -> ( V \ { x } ) = ( V \ { X } ) )
23		simpr 477	. . . . . . . . . . 11 |- ( ( X e. V /\ Y e. V ) -> Y e. V )
24		necom 2847	. . . . . . . . . . . 12 |- ( X =/= Y <-> Y =/= X )
25	24	biimpi 206	. . . . . . . . . . 11 |- ( X =/= Y -> Y =/= X )
26	23, 25	anim12i 590	. . . . . . . . . 10 |- ( ( ( X e. V /\ Y e. V ) /\ X =/= Y ) -> ( Y e. V /\ Y =/= X ) )
27		eldifsn 4317	. . . . . . . . . 10 |- ( Y e. ( V \ { X } ) <-> ( Y e. V /\ Y =/= X ) )
28	26, 27	sylibr 224	. . . . . . . . 9 |- ( ( ( X e. V /\ Y e. V ) /\ X =/= Y ) -> Y e. ( V \ { X } ) )
29	14, 18, 19, 22, 28	rspc2vd 27129	. . . . . . . 8 |- ( ( ( X e. V /\ Y e. V ) /\ X =/= Y ) -> ( A. x e. V A. y e. ( V \ { x } ) ( y e/ ( G NeighbVtx x ) -> ( D ` x ) = ( D ` y ) ) -> ( Y e/ ( G NeighbVtx X ) -> ( D ` X ) = ( D ` Y ) ) ) )
30		nnel 2906	. . . . . . . . . . 11 |- ( -. Y e/ ( G NeighbVtx X ) <-> Y e. ( G NeighbVtx X ) )
31		nbgrsym 26265	. . . . . . . . . . . . . . . 16 |- ( G e. FriendGraph -> ( Y e. ( G NeighbVtx X ) <-> X e. ( G NeighbVtx Y ) ) )
32		frgrusgr 27124	. . . . . . . . . . . . . . . . . 18 |- ( G e. FriendGraph -> G e. USGraph )
33		frgrwopreglem4a.e	. . . . . . . . . . . . . . . . . . 19 |- E = (Edg ` G )
34	33	nbusgreledg 26249	. . . . . . . . . . . . . . . . . 18 |- ( G e. USGraph -> ( X e. ( G NeighbVtx Y ) <-> { X , Y } e. E ) )
35	32, 34	syl 17	. . . . . . . . . . . . . . . . 17 |- ( G e. FriendGraph -> ( X e. ( G NeighbVtx Y ) <-> { X , Y } e. E ) )
36	35	biimpd 219	. . . . . . . . . . . . . . . 16 |- ( G e. FriendGraph -> ( X e. ( G NeighbVtx Y ) -> { X , Y } e. E ) )
37	31, 36	sylbid 230	. . . . . . . . . . . . . . 15 |- ( G e. FriendGraph -> ( Y e. ( G NeighbVtx X ) -> { X , Y } e. E ) )
38	37	imp 445	. . . . . . . . . . . . . 14 |- ( ( G e. FriendGraph /\ Y e. ( G NeighbVtx X ) ) -> { X , Y } e. E )
39	38	a1d 25	. . . . . . . . . . . . 13 |- ( ( G e. FriendGraph /\ Y e. ( G NeighbVtx X ) ) -> ( ( D ` X ) =/= ( D ` Y ) -> { X , Y } e. E ) )
40	39	expcom 451	. . . . . . . . . . . 12 |- ( Y e. ( G NeighbVtx X ) -> ( G e. FriendGraph -> ( ( D ` X ) =/= ( D ` Y ) -> { X , Y } e. E ) ) )
41	40	a1d 25	. . . . . . . . . . 11 |- ( Y e. ( G NeighbVtx X ) -> ( ( ( X e. V /\ Y e. V ) /\ X =/= Y ) -> ( G e. FriendGraph -> ( ( D ` X ) =/= ( D ` Y ) -> { X , Y } e. E ) ) ) )
42	30, 41	sylbi 207	. . . . . . . . . 10 |- ( -. Y e/ ( G NeighbVtx X ) -> ( ( ( X e. V /\ Y e. V ) /\ X =/= Y ) -> ( G e. FriendGraph -> ( ( D ` X ) =/= ( D ` Y ) -> { X , Y } e. E ) ) ) )
43		eqneqall 2805	. . . . . . . . . . 11 |- ( ( D ` X ) = ( D ` Y ) -> ( ( D ` X ) =/= ( D ` Y ) -> { X , Y } e. E ) )
44	43	2a1d 26	. . . . . . . . . 10 |- ( ( D ` X ) = ( D ` Y ) -> ( ( ( X e. V /\ Y e. V ) /\ X =/= Y ) -> ( G e. FriendGraph -> ( ( D ` X ) =/= ( D ` Y ) -> { X , Y } e. E ) ) ) )
45	42, 44	ja 173	. . . . . . . . 9 |- ( ( Y e/ ( G NeighbVtx X ) -> ( D ` X ) = ( D ` Y ) ) -> ( ( ( X e. V /\ Y e. V ) /\ X =/= Y ) -> ( G e. FriendGraph -> ( ( D ` X ) =/= ( D ` Y ) -> { X , Y } e. E ) ) ) )
46	45	com12 32	. . . . . . . 8 |- ( ( ( X e. V /\ Y e. V ) /\ X =/= Y ) -> ( ( Y e/ ( G NeighbVtx X ) -> ( D ` X ) = ( D ` Y ) ) -> ( G e. FriendGraph -> ( ( D ` X ) =/= ( D ` Y ) -> { X , Y } e. E ) ) ) )
47	29, 46	syld 47	. . . . . . 7 |- ( ( ( X e. V /\ Y e. V ) /\ X =/= Y ) -> ( A. x e. V A. y e. ( V \ { x } ) ( y e/ ( G NeighbVtx x ) -> ( D ` x ) = ( D ` y ) ) -> ( G e. FriendGraph -> ( ( D ` X ) =/= ( D ` Y ) -> { X , Y } e. E ) ) ) )
48	47	com3l 89	. . . . . 6 |- ( A. x e. V A. y e. ( V \ { x } ) ( y e/ ( G NeighbVtx x ) -> ( D ` x ) = ( D ` y ) ) -> ( G e. FriendGraph -> ( ( ( X e. V /\ Y e. V ) /\ X =/= Y ) -> ( ( D ` X ) =/= ( D ` Y ) -> { X , Y } e. E ) ) ) )
49	8, 48	mpcom 38	. . . . 5 |- ( G e. FriendGraph -> ( ( ( X e. V /\ Y e. V ) /\ X =/= Y ) -> ( ( D ` X ) =/= ( D ` Y ) -> { X , Y } e. E ) ) )
50	49	expd 452	. . . 4 |- ( G e. FriendGraph -> ( ( X e. V /\ Y e. V ) -> ( X =/= Y -> ( ( D ` X ) =/= ( D ` Y ) -> { X , Y } e. E ) ) ) )
51	50	com34 91	. . 3 |- ( G e. FriendGraph -> ( ( X e. V /\ Y e. V ) -> ( ( D ` X ) =/= ( D ` Y ) -> ( X =/= Y -> { X , Y } e. E ) ) ) )
52	51	3imp 1256	. 2 |- ( ( G e. FriendGraph /\ ( X e. V /\ Y e. V ) /\ ( D ` X ) =/= ( D ` Y ) ) -> ( X =/= Y -> { X , Y } e. E ) )
53	5, 52	mpd 15	1 |- ( ( G e. FriendGraph /\ ( X e. V /\ Y e. V ) /\ ( D ` X ) =/= ( D ` Y ) ) -> { X , Y } e. E )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   e/ wnel 2897   A.wral 2912   \ cdif 3571   {csn 4177   {cpr 4179   ` cfv 5888  (class class class)co 6650  Vtxcvtx 25874  Edgcedg 25939   USGraph cusgr 26044   NeighbVtx cnbgr 26224  VtxDegcvtxdg 26361   FriendGraph cfrgr 27120

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-xadd 11947  df-fz 12327  df-hash 13118  df-edg 25940  df-uhgr 25953  df-ushgr 25954  df-upgr 25977  df-umgr 25978  df-uspgr 26045  df-usgr 26046  df-nbgr 26228  df-vtxdg 26362  df-frgr 27121

This theorem is referenced by:  frgrwopreglem5a  27175  frgrwopreglem4  27179