Theorem umgredg 16069

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 2298	. . . 4 |- ( C e. E <-> C e. (Edg ` G ) )
3		edgumgren 16066	. . . 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 7441	. . . . 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 2294	. . . . . . . . . 10 |- ( C = { a , b } -> ( C e. ~P (Vtx ` G ) <-> { a , b } e. ~P (Vtx ` G ) ) )
8		zfpair2 4306	. . . . . . . . . . . 12 |- { a , b } e. _V
9	8	elpw 3662	. . . . . . . . . . 11 |- ( { a , b } e. ~P (Vtx ` G ) <-> { a , b } C_ (Vtx ` G ) )
10		vex 2806	. . . . . . . . . . . . 13 |- a e. _V
11		vex 2806	. . . . . . . . . . . . 13 |- b e. _V
12	10, 11	prss 3834	. . . . . . . . . . . 12 |- ( ( a e. V /\ b e. V ) <-> { a , b } C_ V )
13		upgredg.v	. . . . . . . . . . . . 13 |- V = (Vtx ` G )
14	13	sseq2i 3255	. . . . . . . . . . . 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 1930	. . . 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 2553	. 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 1398   E.wex 1541   e. wcel 2202   =/= wne 2403   E.wrex 2512   C_ wss 3201   ~Pcpw 3656   {cpr 3674   class class class wbr 4093   ` cfv 5333   2oc2o 6619   ~~ cen 6950  Vtxcvtx 15936  Edgcedg 15981  UMGraphcumgr 16016

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-cnre 8186

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-1o 6625  df-2o 6626  df-er 6745  df-en 6953  df-sub 8394  df-inn 9186  df-2 9244  df-3 9245  df-4 9246  df-5 9247  df-6 9248  df-7 9249  df-8 9250  df-9 9251  df-n0 9445  df-dec 9656  df-ndx 13148  df-slot 13149  df-base 13151  df-edgf 15929  df-vtx 15938  df-iedg 15939  df-edg 15982  df-umgren 16018

This theorem is referenced by:  usgredg  16124