Theorem umgredg 16386

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 2305	. . . 4 |- ( C e. E <-> C e. (Edg ` G ) )
3		edgumgren 16383	. . . 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 7539	. . . . 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 2301	. . . . . . . . . 10 |- ( C = { a , b } -> ( C e. ~P (Vtx ` G ) <-> { a , b } e. ~P (Vtx ` G ) ) )
8		zfpair2 4347	. . . . . . . . . . . 12 |- { a , b } e. _V
9	8	elpw 3694	. . . . . . . . . . 11 |- ( { a , b } e. ~P (Vtx ` G ) <-> { a , b } C_ (Vtx ` G ) )
10		vex 2824	. . . . . . . . . . . . 13 |- a e. _V
11		vex 2824	. . . . . . . . . . . . 13 |- b e. _V
12	10, 11	prss 3871	. . . . . . . . . . . 12 |- ( ( a e. V /\ b e. V ) <-> { a , b } C_ V )
13		upgredg.v	. . . . . . . . . . . . 13 |- V = (Vtx ` G )
14	13	sseq2i 3275	. . . . . . . . . . . 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 450	. . . . . 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 1935	. . . 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 2570	. 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 } ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   E.wex 1545   e. wcel 2209   =/= wne 2420   E.wrex 2529   C_ wss 3220   ~Pcpw 3688   {cpr 3710   class class class wbr 4130   ` cfv 5377   2oc2o 6681   ~~ cen 7020  Vtxcvtx 16253  Edgcedg 16298  UMGraphcumgr 16333

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-1o 6687  df-2o 6688  df-er 6807  df-en 7023  df-sub 8499  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-5 9366  df-6 9367  df-7 9368  df-8 9369  df-9 9370  df-n0 9564  df-dec 9778  df-ndx 13355  df-slot 13356  df-base 13358  df-edgf 16246  df-vtx 16255  df-iedg 16256  df-edg 16299  df-umgren 16335

This theorem is used by:  usgredg  16441