Theorem usgraedgrnv 23447

Description: An edge of an undirected simple graph always connects two vertices. (Contributed by Alexander van der Vekens, 7-Oct-2017.)

Assertion

Ref	Expression
usgraedgrnv	|- ( ( V USGrph E /\ { M , N } e. ran E ) -> ( M e. V /\ N e. V ) )

Proof of Theorem usgraedgrnv

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		usgraf0 23427	. . 3 |- ( V USGrph E -> E : dom E -1-1-> { x e. ~P V | ( # ` x ) = 2 } )
2		f1f 5713	. . 3 |- ( E : dom E -1-1-> { x e. ~P V | ( # ` x ) = 2 } -> E : dom E --> { x e. ~P V | ( # ` x ) = 2 } )
3		df-f 5529	. . . 4 |- ( E : dom E --> { x e. ~P V | ( # ` x ) = 2 } <-> ( E Fn dom E /\ ran E C_ { x e. ~P V | ( # ` x ) = 2 } ) )
4		ssel2 3458	. . . . . . 7 |- ( ( ran E C_ { x e. ~P V | ( # ` x ) = 2 } /\ { M , N } e. ran E ) -> { M , N } e. { x e. ~P V | ( # ` x ) = 2 } )
5		fveq2 5798	. . . . . . . . . 10 |- ( x = { M , N } -> ( # ` x ) = ( # ` { M , N } ) )
6	5	eqeq1d 2456	. . . . . . . . 9 |- ( x = { M , N } -> ( ( # ` x ) = 2 <-> ( # ` { M , N } ) = 2 ) )
7	6	elrab 3222	. . . . . . . 8 |- ( { M , N } e. { x e. ~P V | ( # ` x ) = 2 } <-> ( { M , N } e. ~P V /\ ( # ` { M , N } ) = 2 ) )
8		prex 4641	. . . . . . . . . . 11 |- { M , N } e. _V
9	8	elpw 3973	. . . . . . . . . 10 |- ( { M , N } e. ~P V <-> { M , N } C_ V )
10		ianor 488	. . . . . . . . . . . . . 14 |- ( -. ( M e. _V /\ N e. _V ) <-> ( -. M e. _V \/ -. N e. _V ) )
11		elprchashprn2 12273	. . . . . . . . . . . . . . 15 |- ( -. M e. _V -> -. ( # ` { M , N } ) = 2 )
12		elprchashprn2 12273	. . . . . . . . . . . . . . . 16 |- ( -. N e. _V -> -. ( # ` { N , M } ) = 2 )
13		prcom 4060	. . . . . . . . . . . . . . . . . . 19 |- { N , M } = { M , N }
14	13	a1i 11	. . . . . . . . . . . . . . . . . 18 |- ( -. N e. _V -> { N , M } = { M , N } )
15	14	fveq2d 5802	. . . . . . . . . . . . . . . . 17 |- ( -. N e. _V -> ( # ` { N , M } ) = ( # ` { M , N } ) )
16	15	eqeq1d 2456	. . . . . . . . . . . . . . . 16 |- ( -. N e. _V -> ( ( # ` { N , M } ) = 2 <-> ( # ` { M , N } ) = 2 ) )
17	12, 16	mtbid 300	. . . . . . . . . . . . . . 15 |- ( -. N e. _V -> -. ( # ` { M , N } ) = 2 )
18	11, 17	jaoi 379	. . . . . . . . . . . . . 14 |- ( ( -. M e. _V \/ -. N e. _V ) -> -. ( # ` { M , N } ) = 2 )
19	10, 18	sylbi 195	. . . . . . . . . . . . 13 |- ( -. ( M e. _V /\ N e. _V ) -> -. ( # ` { M , N } ) = 2 )
20	19	con4i 130	. . . . . . . . . . . 12 |- ( ( # ` { M , N } ) = 2 -> ( M e. _V /\ N e. _V ) )
21		prssg 4135	. . . . . . . . . . . . 13 |- ( ( M e. _V /\ N e. _V ) -> ( ( M e. V /\ N e. V ) <-> { M , N } C_ V ) )
22	21	biimprd 223	. . . . . . . . . . . 12 |- ( ( M e. _V /\ N e. _V ) -> ( { M , N } C_ V -> ( M e. V /\ N e. V ) ) )
23	20, 22	syl 16	. . . . . . . . . . 11 |- ( ( # ` { M , N } ) = 2 -> ( { M , N } C_ V -> ( M e. V /\ N e. V ) ) )
24	23	com12 31	. . . . . . . . . 10 |- ( { M , N } C_ V -> ( ( # ` { M , N } ) = 2 -> ( M e. V /\ N e. V ) ) )
25	9, 24	sylbi 195	. . . . . . . . 9 |- ( { M , N } e. ~P V -> ( ( # ` { M , N } ) = 2 -> ( M e. V /\ N e. V ) ) )
26	25	imp 429	. . . . . . . 8 |- ( ( { M , N } e. ~P V /\ ( # ` { M , N } ) = 2 ) -> ( M e. V /\ N e. V ) )
27	7, 26	sylbi 195	. . . . . . 7 |- ( { M , N } e. { x e. ~P V | ( # ` x ) = 2 } -> ( M e. V /\ N e. V ) )
28	4, 27	syl 16	. . . . . 6 |- ( ( ran E C_ { x e. ~P V | ( # ` x ) = 2 } /\ { M , N } e. ran E ) -> ( M e. V /\ N e. V ) )
29	28	ex 434	. . . . 5 |- ( ran E C_ { x e. ~P V | ( # ` x ) = 2 } -> ( { M , N } e. ran E -> ( M e. V /\ N e. V ) ) )
30	29	adantl 466	. . . 4 |- ( ( E Fn dom E /\ ran E C_ { x e. ~P V | ( # ` x ) = 2 } ) -> ( { M , N } e. ran E -> ( M e. V /\ N e. V ) ) )
31	3, 30	sylbi 195	. . 3 |- ( E : dom E --> { x e. ~P V | ( # ` x ) = 2 } -> ( { M , N } e. ran E -> ( M e. V /\ N e. V ) ) )
32	1, 2, 31	3syl 20	. 2 |- ( V USGrph E -> ( { M , N } e. ran E -> ( M e. V /\ N e. V ) ) )
33	32	imp 429	1 |- ( ( V USGrph E /\ { M , N } e. ran E ) -> ( M e. V /\ N e. V ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 368   /\ wa 369   = wceq 1370   e. wcel 1758   {crab 2802   _Vcvv 3076   C_ wss 3435   ~Pcpw 3967   {cpr 3986   class class class wbr 4399   dom cdm 4947   ran crn 4948   Fn wfn 5520   -->wf 5521   -1-1->wf1 5522   ` cfv 5525   2c2 10481   #chash 12219   USGrph cusg 23415

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-om 6586  df-1st 6686  df-2nd 6687  df-recs 6941  df-rdg 6975  df-1o 7029  df-er 7210  df-en 7420  df-dom 7421  df-sdom 7422  df-fin 7423  df-card 8219  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-nn 10433  df-2 10490  df-n0 10690  df-z 10757  df-uz 10972  df-fz 11554  df-hash 12220  df-usgra 23417

This theorem is referenced by:  nbusgra  23490  nbgra0nb  23491  nbgraeledg  23492  nbgraisvtx  23493  constr3trllem2  23688  constr3trllem5  23691  usgra2adedgspthlem2  30451  usgra2adedgspth  30452  usgra2adedgwlk  30453  usgra2adedgwlkon  30454  frgranbnb  30759  frgraeu  30794  extwwlkfablem1  30814  numclwwlkun  30819