Theorem umgrvad2edg 16255

Description: If a vertex is adjacent to two different vertices in a multigraph, there are more than one edges starting at this vertex, analogous to usgr2edg 16252. (Contributed by Alexander van der Vekens, 10-Dec-2017.) (Revised by AV, 9-Jan-2020.) (Revised by AV, 8-Jun-2021.)

Hypothesis

Ref	Expression
umgrvad2edg.e	|- E = (Edg ` G )

Assertion

Ref	Expression
umgrvad2edg	|- ( ( ( G e. UMGraph /\ A =/= B ) /\ ( { N , A } e. E /\ { B , N } e. E ) ) -> E. x e. E E. y e. E ( x =/= y /\ N e. x /\ N e. y ) )

Distinct variable groups:   x, A, y   x, B, y   x, E, y   x, G, y   x, N, y

Proof of Theorem umgrvad2edg

Step	Hyp	Ref	Expression
1		simpl 109	. 2 |- ( ( { N , A } e. E /\ { B , N } e. E ) -> { N , A } e. E )
2		simpr 110	. 2 |- ( ( { N , A } e. E /\ { B , N } e. E ) -> { B , N } e. E )
3		eqid 2234	. . . . . . . 8 |- (Vtx ` G ) = (Vtx ` G )
4		umgrvad2edg.e	. . . . . . . 8 |- E = (Edg ` G )
5	3, 4	umgrpredgv 16191	. . . . . . 7 |- ( ( G e. UMGraph /\ { N , A } e. E ) -> ( N e. (Vtx ` G ) /\ A e. (Vtx ` G ) ) )
6	5	ex 115	. . . . . 6 |- ( G e. UMGraph -> ( { N , A } e. E -> ( N e. (Vtx ` G ) /\ A e. (Vtx ` G ) ) ) )
7	3, 4	umgrpredgv 16191	. . . . . . 7 |- ( ( G e. UMGraph /\ { B , N } e. E ) -> ( B e. (Vtx ` G ) /\ N e. (Vtx ` G ) ) )
8	7	ex 115	. . . . . 6 |- ( G e. UMGraph -> ( { B , N } e. E -> ( B e. (Vtx ` G ) /\ N e. (Vtx ` G ) ) ) )
9	6, 8	anim12d 335	. . . . 5 |- ( G e. UMGraph -> ( ( { N , A } e. E /\ { B , N } e. E ) -> ( ( N e. (Vtx ` G ) /\ A e. (Vtx ` G ) ) /\ ( B e. (Vtx ` G ) /\ N e. (Vtx ` G ) ) ) ) )
10	9	adantr 276	. . . 4 |- ( ( G e. UMGraph /\ A =/= B ) -> ( ( { N , A } e. E /\ { B , N } e. E ) -> ( ( N e. (Vtx ` G ) /\ A e. (Vtx ` G ) ) /\ ( B e. (Vtx ` G ) /\ N e. (Vtx ` G ) ) ) ) )
11	10	imp 124	. . 3 |- ( ( ( G e. UMGraph /\ A =/= B ) /\ ( { N , A } e. E /\ { B , N } e. E ) ) -> ( ( N e. (Vtx ` G ) /\ A e. (Vtx ` G ) ) /\ ( B e. (Vtx ` G ) /\ N e. (Vtx ` G ) ) ) )
12		simplr 529	. . . . 5 |- ( ( ( G e. UMGraph /\ A =/= B ) /\ ( { N , A } e. E /\ { B , N } e. E ) ) -> A =/= B )
13	4	umgredgne 16194	. . . . . . 7 |- ( ( G e. UMGraph /\ { N , A } e. E ) -> N =/= A )
14	13	necomd 2500	. . . . . 6 |- ( ( G e. UMGraph /\ { N , A } e. E ) -> A =/= N )
15	14	ad2ant2r 509	. . . . 5 |- ( ( ( G e. UMGraph /\ A =/= B ) /\ ( { N , A } e. E /\ { B , N } e. E ) ) -> A =/= N )
16	12, 15	jca 306	. . . 4 |- ( ( ( G e. UMGraph /\ A =/= B ) /\ ( { N , A } e. E /\ { B , N } e. E ) ) -> ( A =/= B /\ A =/= N ) )
17	16	olcd 742	. . 3 |- ( ( ( G e. UMGraph /\ A =/= B ) /\ ( { N , A } e. E /\ { B , N } e. E ) ) -> ( ( N =/= B /\ N =/= N ) \/ ( A =/= B /\ A =/= N ) ) )
18		prneimg 3880	. . . . 5 |- ( ( ( N e. (Vtx ` G ) /\ A e. (Vtx ` G ) ) /\ ( B e. (Vtx ` G ) /\ N e. (Vtx ` G ) ) ) -> ( ( ( N =/= B /\ N =/= N ) \/ ( A =/= B /\ A =/= N ) ) -> { N , A } =/= { B , N } ) )
19	18	imp 124	. . . 4 |- ( ( ( ( N e. (Vtx ` G ) /\ A e. (Vtx ` G ) ) /\ ( B e. (Vtx ` G ) /\ N e. (Vtx ` G ) ) ) /\ ( ( N =/= B /\ N =/= N ) \/ ( A =/= B /\ A =/= N ) ) ) -> { N , A } =/= { B , N } )
20		prid1g 3797	. . . . 5 |- ( N e. (Vtx ` G ) -> N e. { N , A } )
21	20	ad3antrrr 492	. . . 4 |- ( ( ( ( N e. (Vtx ` G ) /\ A e. (Vtx ` G ) ) /\ ( B e. (Vtx ` G ) /\ N e. (Vtx ` G ) ) ) /\ ( ( N =/= B /\ N =/= N ) \/ ( A =/= B /\ A =/= N ) ) ) -> N e. { N , A } )
22		prid2g 3798	. . . . 5 |- ( N e. (Vtx ` G ) -> N e. { B , N } )
23	22	ad3antrrr 492	. . . 4 |- ( ( ( ( N e. (Vtx ` G ) /\ A e. (Vtx ` G ) ) /\ ( B e. (Vtx ` G ) /\ N e. (Vtx ` G ) ) ) /\ ( ( N =/= B /\ N =/= N ) \/ ( A =/= B /\ A =/= N ) ) ) -> N e. { B , N } )
24	19, 21, 23	3jca 1204	. . 3 |- ( ( ( ( N e. (Vtx ` G ) /\ A e. (Vtx ` G ) ) /\ ( B e. (Vtx ` G ) /\ N e. (Vtx ` G ) ) ) /\ ( ( N =/= B /\ N =/= N ) \/ ( A =/= B /\ A =/= N ) ) ) -> ( { N , A } =/= { B , N } /\ N e. { N , A } /\ N e. { B , N } ) )
25	11, 17, 24	syl2anc 411	. 2 |- ( ( ( G e. UMGraph /\ A =/= B ) /\ ( { N , A } e. E /\ { B , N } e. E ) ) -> ( { N , A } =/= { B , N } /\ N e. { N , A } /\ N e. { B , N } ) )
26		neeq1 2427	. . . 4 |- ( x = { N , A } -> ( x =/= y <-> { N , A } =/= y ) )
27		eleq2 2298	. . . 4 |- ( x = { N , A } -> ( N e. x <-> N e. { N , A } ) )
28	26, 27	3anbi12d 1350	. . 3 |- ( x = { N , A } -> ( ( x =/= y /\ N e. x /\ N e. y ) <-> ( { N , A } =/= y /\ N e. { N , A } /\ N e. y ) ) )
29		neeq2 2428	. . . 4 |- ( y = { B , N } -> ( { N , A } =/= y <-> { N , A } =/= { B , N } ) )
30		eleq2 2298	. . . 4 |- ( y = { B , N } -> ( N e. y <-> N e. { B , N } ) )
31	29, 30	3anbi13d 1351	. . 3 |- ( y = { B , N } -> ( ( { N , A } =/= y /\ N e. { N , A } /\ N e. y ) <-> ( { N , A } =/= { B , N } /\ N e. { N , A } /\ N e. { B , N } ) ) )
32	28, 31	rspc2ev 2938	. 2 |- ( ( { N , A } e. E /\ { B , N } e. E /\ ( { N , A } =/= { B , N } /\ N e. { N , A } /\ N e. { B , N } ) ) -> E. x e. E E. y e. E ( x =/= y /\ N e. x /\ N e. y ) )
33	1, 2, 25, 32	syl2an23an 1336	1 |- ( ( ( G e. UMGraph /\ A =/= B ) /\ ( { N , A } e. E /\ { B , N } e. E ) ) -> E. x e. E E. y e. E ( x =/= y /\ N e. x /\ N e. y ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 716   /\ w3a 1005   = wceq 1398   e. wcel 2205   =/= wne 2414   E.wrex 2523   {cpr 3692   ` cfv 5354  Vtxcvtx 16056  Edgcedg 16101  UMGraphcumgr 16136

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  ax-cnex 8223  ax-resscn 8224  ax-1cn 8225  ax-1re 8226  ax-icn 8227  ax-addcl 8228  ax-addrcl 8229  ax-mulcl 8230  ax-addcom 8232  ax-mulcom 8233  ax-addass 8234  ax-mulass 8235  ax-distr 8236  ax-i2m1 8237  ax-1rid 8239  ax-0id 8240  ax-rnegex 8241  ax-cnre 8243

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-iord 4489  df-on 4491  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-1o 6649  df-2o 6650  df-er 6769  df-en 6978  df-sub 8451  df-inn 9243  df-2 9301  df-3 9302  df-4 9303  df-5 9304  df-6 9305  df-7 9306  df-8 9307  df-9 9308  df-n0 9502  df-dec 9716  df-ndx 13236  df-slot 13237  df-base 13239  df-edgf 16049  df-vtx 16058  df-iedg 16059  df-edg 16102  df-umgren 16138

This theorem is referenced by:  umgr2edgneu  16256