Theorem umgrvad2edg 16091

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 16088. (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 2230	. . . . . . . 8 |- (Vtx ` G ) = (Vtx ` G )
4		umgrvad2edg.e	. . . . . . . 8 |- E = (Edg ` G )
5	3, 4	umgrpredgv 16027	. . . . . . 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 16027	. . . . . . 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 16030	. . . . . . 7 |- ( ( G e. UMGraph /\ { N , A } e. E ) -> N =/= A )
14	13	necomd 2487	. . . . . 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 741	. . 3 |- ( ( ( G e. UMGraph /\ A =/= B ) /\ ( { N , A } e. E /\ { B , N } e. E ) ) -> ( ( N =/= B /\ N =/= N ) \/ ( A =/= B /\ A =/= N ) ) )
18		prneimg 3858	. . . . 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 3776	. . . . 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 3777	. . . . 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 1203	. . 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 2414	. . . 4 |- ( x = { N , A } -> ( x =/= y <-> { N , A } =/= y ) )
27		eleq2 2294	. . . 4 |- ( x = { N , A } -> ( N e. x <-> N e. { N , A } ) )
28	26, 27	3anbi12d 1349	. . 3 |- ( x = { N , A } -> ( ( x =/= y /\ N e. x /\ N e. y ) <-> ( { N , A } =/= y /\ N e. { N , A } /\ N e. y ) ) )
29		neeq2 2415	. . . 4 |- ( y = { B , N } -> ( { N , A } =/= y <-> { N , A } =/= { B , N } ) )
30		eleq2 2294	. . . 4 |- ( y = { B , N } -> ( N e. y <-> N e. { B , N } ) )
31	29, 30	3anbi13d 1350	. . 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 2924	. 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 1335	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 715   /\ w3a 1004   = wceq 1397   e. wcel 2201   =/= wne 2401   E.wrex 2510   {cpr 3671   ` cfv 5328  Vtxcvtx 15892  Edgcedg 15937  UMGraphcumgr 15972

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-nul 4216  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-setind 4637  ax-iinf 4688  ax-cnex 8128  ax-resscn 8129  ax-1cn 8130  ax-1re 8131  ax-icn 8132  ax-addcl 8133  ax-addrcl 8134  ax-mulcl 8135  ax-addcom 8137  ax-mulcom 8138  ax-addass 8139  ax-mulass 8140  ax-distr 8141  ax-i2m1 8142  ax-1rid 8144  ax-0id 8145  ax-rnegex 8146  ax-cnre 8148

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-ral 2514  df-rex 2515  df-reu 2516  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-if 3605  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-int 3930  df-br 4090  df-opab 4152  df-mpt 4153  df-tr 4189  df-id 4392  df-iord 4465  df-on 4467  df-suc 4470  df-iom 4691  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-ima 4740  df-iota 5288  df-fun 5330  df-fn 5331  df-f 5332  df-f1 5333  df-fo 5334  df-f1o 5335  df-fv 5336  df-riota 5976  df-ov 6026  df-oprab 6027  df-mpo 6028  df-1st 6308  df-2nd 6309  df-1o 6587  df-2o 6588  df-er 6707  df-en 6915  df-sub 8357  df-inn 9149  df-2 9207  df-3 9208  df-4 9209  df-5 9210  df-6 9211  df-7 9212  df-8 9213  df-9 9214  df-n0 9408  df-dec 9617  df-ndx 13108  df-slot 13109  df-base 13111  df-edgf 15885  df-vtx 15894  df-iedg 15895  df-edg 15938  df-umgren 15974

This theorem is referenced by:  umgr2edgneu  16092