Theorem frgrwopreglem5a 27175

Description: If a friendship graph has two vertices with the same degree and two other vertices with different degrees, then there is a 4-cycle in the graph. Alternate version of frgrwopreglem5 27185 without a fixed degree and without using the sets A and B. (Contributed by Alexander van der Vekens, 31-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
frgrwopreglem5a	|- ( ( G e. FriendGraph /\ ( ( A e. V /\ X e. V ) /\ ( B e. V /\ Y e. V ) ) /\ ( ( D ` A ) = ( D ` X ) /\ ( D ` A ) =/= ( D ` B ) /\ ( D ` X ) =/= ( D ` Y ) ) ) -> ( ( { A , B } e. E /\ { B , X } e. E ) /\ ( { X , Y } e. E /\ { Y , A } e. E ) ) )

Proof of Theorem frgrwopreglem5a

Step	Hyp	Ref	Expression
1		id 22	. . 3 |- ( G e. FriendGraph -> G e. FriendGraph )
2		simpl 473	. . . 4 |- ( ( A e. V /\ X e. V ) -> A e. V )
3		simpl 473	. . . 4 |- ( ( B e. V /\ Y e. V ) -> B e. V )
4	2, 3	anim12i 590	. . 3 |- ( ( ( A e. V /\ X e. V ) /\ ( B e. V /\ Y e. V ) ) -> ( A e. V /\ B e. V ) )
5		simp2 1062	. . 3 |- ( ( ( D ` A ) = ( D ` X ) /\ ( D ` A ) =/= ( D ` B ) /\ ( D ` X ) =/= ( D ` Y ) ) -> ( D ` A ) =/= ( D ` B ) )
6		frgrncvvdeq.v	. . . 4 |- V = (Vtx ` G )
7		frgrncvvdeq.d	. . . 4 |- D = (VtxDeg ` G )
8		frgrwopreglem4a.e	. . . 4 |- E = (Edg ` G )
9	6, 7, 8	frgrwopreglem4a 27174	. . 3 |- ( ( G e. FriendGraph /\ ( A e. V /\ B e. V ) /\ ( D ` A ) =/= ( D ` B ) ) -> { A , B } e. E )
10	1, 4, 5, 9	syl3an 1368	. 2 |- ( ( G e. FriendGraph /\ ( ( A e. V /\ X e. V ) /\ ( B e. V /\ Y e. V ) ) /\ ( ( D ` A ) = ( D ` X ) /\ ( D ` A ) =/= ( D ` B ) /\ ( D ` X ) =/= ( D ` Y ) ) ) -> { A , B } e. E )
11		simpr 477	. . . 4 |- ( ( A e. V /\ X e. V ) -> X e. V )
12	11, 3	anim12ci 591	. . 3 |- ( ( ( A e. V /\ X e. V ) /\ ( B e. V /\ Y e. V ) ) -> ( B e. V /\ X e. V ) )
13		pm13.18 2875	. . . . 5 |- ( ( ( D ` A ) = ( D ` X ) /\ ( D ` A ) =/= ( D ` B ) ) -> ( D ` X ) =/= ( D ` B ) )
14	13	3adant3 1081	. . . 4 |- ( ( ( D ` A ) = ( D ` X ) /\ ( D ` A ) =/= ( D ` B ) /\ ( D ` X ) =/= ( D ` Y ) ) -> ( D ` X ) =/= ( D ` B ) )
15	14	necomd 2849	. . 3 |- ( ( ( D ` A ) = ( D ` X ) /\ ( D ` A ) =/= ( D ` B ) /\ ( D ` X ) =/= ( D ` Y ) ) -> ( D ` B ) =/= ( D ` X ) )
16	6, 7, 8	frgrwopreglem4a 27174	. . 3 |- ( ( G e. FriendGraph /\ ( B e. V /\ X e. V ) /\ ( D ` B ) =/= ( D ` X ) ) -> { B , X } e. E )
17	1, 12, 15, 16	syl3an 1368	. 2 |- ( ( G e. FriendGraph /\ ( ( A e. V /\ X e. V ) /\ ( B e. V /\ Y e. V ) ) /\ ( ( D ` A ) = ( D ` X ) /\ ( D ` A ) =/= ( D ` B ) /\ ( D ` X ) =/= ( D ` Y ) ) ) -> { B , X } e. E )
18		simpr 477	. . . . 5 |- ( ( B e. V /\ Y e. V ) -> Y e. V )
19	11, 18	anim12i 590	. . . 4 |- ( ( ( A e. V /\ X e. V ) /\ ( B e. V /\ Y e. V ) ) -> ( X e. V /\ Y e. V ) )
20		simp3 1063	. . . 4 |- ( ( ( D ` A ) = ( D ` X ) /\ ( D ` A ) =/= ( D ` B ) /\ ( D ` X ) =/= ( D ` Y ) ) -> ( D ` X ) =/= ( D ` Y ) )
21	6, 7, 8	frgrwopreglem4a 27174	. . . 4 |- ( ( G e. FriendGraph /\ ( X e. V /\ Y e. V ) /\ ( D ` X ) =/= ( D ` Y ) ) -> { X , Y } e. E )
22	1, 19, 20, 21	syl3an 1368	. . 3 |- ( ( G e. FriendGraph /\ ( ( A e. V /\ X e. V ) /\ ( B e. V /\ Y e. V ) ) /\ ( ( D ` A ) = ( D ` X ) /\ ( D ` A ) =/= ( D ` B ) /\ ( D ` X ) =/= ( D ` Y ) ) ) -> { X , Y } e. E )
23	2, 18	anim12ci 591	. . . 4 |- ( ( ( A e. V /\ X e. V ) /\ ( B e. V /\ Y e. V ) ) -> ( Y e. V /\ A e. V ) )
24		pm13.181 2876	. . . . . 6 |- ( ( ( D ` A ) = ( D ` X ) /\ ( D ` X ) =/= ( D ` Y ) ) -> ( D ` A ) =/= ( D ` Y ) )
25	24	3adant2 1080	. . . . 5 |- ( ( ( D ` A ) = ( D ` X ) /\ ( D ` A ) =/= ( D ` B ) /\ ( D ` X ) =/= ( D ` Y ) ) -> ( D ` A ) =/= ( D ` Y ) )
26	25	necomd 2849	. . . 4 |- ( ( ( D ` A ) = ( D ` X ) /\ ( D ` A ) =/= ( D ` B ) /\ ( D ` X ) =/= ( D ` Y ) ) -> ( D ` Y ) =/= ( D ` A ) )
27	6, 7, 8	frgrwopreglem4a 27174	. . . 4 |- ( ( G e. FriendGraph /\ ( Y e. V /\ A e. V ) /\ ( D ` Y ) =/= ( D ` A ) ) -> { Y , A } e. E )
28	1, 23, 26, 27	syl3an 1368	. . 3 |- ( ( G e. FriendGraph /\ ( ( A e. V /\ X e. V ) /\ ( B e. V /\ Y e. V ) ) /\ ( ( D ` A ) = ( D ` X ) /\ ( D ` A ) =/= ( D ` B ) /\ ( D ` X ) =/= ( D ` Y ) ) ) -> { Y , A } e. E )
29	22, 28	jca 554	. 2 |- ( ( G e. FriendGraph /\ ( ( A e. V /\ X e. V ) /\ ( B e. V /\ Y e. V ) ) /\ ( ( D ` A ) = ( D ` X ) /\ ( D ` A ) =/= ( D ` B ) /\ ( D ` X ) =/= ( D ` Y ) ) ) -> ( { X , Y } e. E /\ { Y , A } e. E ) )
30	10, 17, 29	jca31 557	1 |- ( ( G e. FriendGraph /\ ( ( A e. V /\ X e. V ) /\ ( B e. V /\ Y e. V ) ) /\ ( ( D ` A ) = ( D ` X ) /\ ( D ` A ) =/= ( D ` B ) /\ ( D ` X ) =/= ( D ` Y ) ) ) -> ( ( { A , B } e. E /\ { B , X } e. E ) /\ ( { X , Y } e. E /\ { Y , A } e. E ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   {cpr 4179   ` cfv 5888  Vtxcvtx 25874  Edgcedg 25939  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:  frgrwopreglem5  27185