Theorem umgredg 15958

Description: For each edge in a multigraph, there are two distinct vertices which are connected by this edge. (Contributed by Alexander van der Vekens, 9-Dec-2017.) (Revised by AV, 25-Nov-2020.)

Hypotheses

Ref	Expression
upgredg.v	|- V = (Vtx ` G )
upgredg.e	|- E = (Edg ` G )

Assertion

Ref	Expression
umgredg	|- ( ( G e. UMGraph /\ C e. E ) -> E. a e. V E. b e. V ( a =/= b /\ C = { a , b } ) )

Distinct variable groups:   C, a, b   G, a, b   V, a, b

Allowed substitution hints:   E( a, b)

Proof of Theorem umgredg

Step	Hyp	Ref	Expression
1		upgredg.e	. . . . 5 |- E = (Edg ` G )
2	1	eleq2i 2296	. . . 4 |- ( C e. E <-> C e. (Edg ` G ) )
3		edgumgren 15955	. . . 4 |- ( ( G e. UMGraph /\ C e. (Edg ` G ) ) -> ( C e. ~P (Vtx ` G ) /\ C ~~ 2o ) )
4	2, 3	sylan2b 287	. . 3 |- ( ( G e. UMGraph /\ C e. E ) -> ( C e. ~P (Vtx ` G ) /\ C ~~ 2o ) )
5		en2prde 7377	. . . . 5 |- ( C ~~ 2o -> E. a E. b ( a =/= b /\ C = { a , b } ) )
6	5	adantl 277	. . . 4 |- ( ( C e. ~P (Vtx ` G ) /\ C ~~ 2o ) -> E. a E. b ( a =/= b /\ C = { a , b } ) )
7		eleq1 2292	. . . . . . . . . 10 |- ( C = { a , b } -> ( C e. ~P (Vtx ` G ) <-> { a , b } e. ~P (Vtx ` G ) ) )
8		zfpair2 4294	. . . . . . . . . . . 12 |- { a , b } e. _V
9	8	elpw 3655	. . . . . . . . . . 11 |- ( { a , b } e. ~P (Vtx ` G ) <-> { a , b } C_ (Vtx ` G ) )
10		vex 2802	. . . . . . . . . . . . 13 |- a e. _V
11		vex 2802	. . . . . . . . . . . . 13 |- b e. _V
12	10, 11	prss 3824	. . . . . . . . . . . 12 |- ( ( a e. V /\ b e. V ) <-> { a , b } C_ V )
13		upgredg.v	. . . . . . . . . . . . 13 |- V = (Vtx ` G )
14	13	sseq2i 3251	. . . . . . . . . . . 12 |- ( { a , b } C_ V <-> { a , b } C_ (Vtx ` G ) )
15	12, 14	sylbbr 136	. . . . . . . . . . 11 |- ( { a , b } C_ (Vtx ` G ) -> ( a e. V /\ b e. V ) )
16	9, 15	sylbi 121	. . . . . . . . . 10 |- ( { a , b } e. ~P (Vtx ` G ) -> ( a e. V /\ b e. V ) )
17	7, 16	biimtrdi 163	. . . . . . . . 9 |- ( C = { a , b } -> ( C e. ~P (Vtx ` G ) -> ( a e. V /\ b e. V ) ) )
18	17	adantrd 279	. . . . . . . 8 |- ( C = { a , b } -> ( ( C e. ~P (Vtx ` G ) /\ C ~~ 2o ) -> ( a e. V /\ b e. V ) ) )
19	18	adantl 277	. . . . . . 7 |- ( ( a =/= b /\ C = { a , b } ) -> ( ( C e. ~P (Vtx ` G ) /\ C ~~ 2o ) -> ( a e. V /\ b e. V ) ) )
20	19	imdistanri 446	. . . . . 6 |- ( ( ( C e. ~P (Vtx ` G ) /\ C ~~ 2o ) /\ ( a =/= b /\ C = { a , b } ) ) -> ( ( a e. V /\ b e. V ) /\ ( a =/= b /\ C = { a , b } ) ) )
21	20	ex 115	. . . . 5 |- ( ( C e. ~P (Vtx ` G ) /\ C ~~ 2o ) -> ( ( a =/= b /\ C = { a , b } ) -> ( ( a e. V /\ b e. V ) /\ ( a =/= b /\ C = { a , b } ) ) ) )
22	21	2eximdv 1928	. . . 4 |- ( ( C e. ~P (Vtx ` G ) /\ C ~~ 2o ) -> ( E. a E. b ( a =/= b /\ C = { a , b } ) -> E. a E. b ( ( a e. V /\ b e. V ) /\ ( a =/= b /\ C = { a , b } ) ) ) )
23	6, 22	mpd 13	. . 3 |- ( ( C e. ~P (Vtx ` G ) /\ C ~~ 2o ) -> E. a E. b ( ( a e. V /\ b e. V ) /\ ( a =/= b /\ C = { a , b } ) ) )
24	4, 23	syl 14	. 2 |- ( ( G e. UMGraph /\ C e. E ) -> E. a E. b ( ( a e. V /\ b e. V ) /\ ( a =/= b /\ C = { a , b } ) ) )
25		r2ex 2550	. 2 |- ( E. a e. V E. b e. V ( a =/= b /\ C = { a , b } ) <-> E. a E. b ( ( a e. V /\ b e. V ) /\ ( a =/= b /\ C = { a , b } ) ) )
26	24, 25	sylibr 134	1 |- ( ( G e. UMGraph /\ C e. E ) -> E. a e. V E. b e. V ( a =/= b /\ C = { a , b } ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   E.wex 1538   e. wcel 2200   =/= wne 2400   E.wrex 2509   C_ wss 3197   ~Pcpw 3649   {cpr 3667   class class class wbr 4083   ` cfv 5318   2oc2o 6562   ~~ cen 6893  Vtxcvtx 15828  Edgcedg 15873  UMGraphcumgr 15907

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-addcom 8110  ax-mulcom 8111  ax-addass 8112  ax-mulass 8113  ax-distr 8114  ax-i2m1 8115  ax-1rid 8117  ax-0id 8118  ax-rnegex 8119  ax-cnre 8121

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-1o 6568  df-2o 6569  df-er 6688  df-en 6896  df-sub 8330  df-inn 9122  df-2 9180  df-3 9181  df-4 9182  df-5 9183  df-6 9184  df-7 9185  df-8 9186  df-9 9187  df-n0 9381  df-dec 9590  df-ndx 13050  df-slot 13051  df-base 13053  df-edgf 15821  df-vtx 15830  df-iedg 15831  df-edg 15874  df-umgren 15909

This theorem is referenced by:  usgredg  16013