Theorem nbgra0nb 25169

Description: A vertex which is not endpoint of an edge has no neighbor. (Contributed by Alexander van der Vekens, 12-Oct-2017.)

Assertion

Ref	Expression
nbgra0nb	|- ( V USGrph E -> ( A. x e. ran E N e/ x -> ( <. V , E >. Neighbors N ) = (/) ) )

Distinct variable groups:   x, V   x, E   x, N

Proof of Theorem nbgra0nb

Dummy variable n is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nbusgra 25168	. . . 4 |- ( V USGrph E -> ( <. V , E >. Neighbors N ) = { n e. V | { N , n } e. ran E } )
2	1	adantr 467	. . 3 |- ( ( V USGrph E /\ A. x e. ran E N e/ x ) -> ( <. V , E >. Neighbors N ) = { n e. V | { N , n } e. ran E } )
3		neleq2 2732	. . . . . . . . . 10 |- ( x = { N , n } -> ( N e/ x <-> N e/ { N , n } ) )
4	3	rspcv 3148	. . . . . . . . 9 |- ( { N , n } e. ran E -> ( A. x e. ran E N e/ x -> N e/ { N , n } ) )
5		df-nel 2627	. . . . . . . . . 10 |- ( N e/ { N , n } <-> -. N e. { N , n } )
6		elprg 3986	. . . . . . . . . . . . . 14 |- ( N e. V -> ( N e. { N , n } <-> ( N = N \/ N = n ) ) )
7	6	notbid 296	. . . . . . . . . . . . 13 |- ( N e. V -> ( -. N e. { N , n } <-> -. ( N = N \/ N = n ) ) )
8		ioran 493	. . . . . . . . . . . . 13 |- ( -. ( N = N \/ N = n ) <-> ( -. N = N /\ -. N = n ) )
9	7, 8	syl6bb 265	. . . . . . . . . . . 12 |- ( N e. V -> ( -. N e. { N , n } <-> ( -. N = N /\ -. N = n ) ) )
10		eqid 2453	. . . . . . . . . . . . . 14 |- N = N
11	10	pm2.24i 137	. . . . . . . . . . . . 13 |- ( -. N = N -> ( -. N = n -> ( V USGrph E -> ( n e. V -> -. { N , n } e. ran E ) ) ) )
12	11	imp 431	. . . . . . . . . . . 12 |- ( ( -. N = N /\ -. N = n ) -> ( V USGrph E -> ( n e. V -> -. { N , n } e. ran E ) ) )
13	9, 12	syl6bi 232	. . . . . . . . . . 11 |- ( N e. V -> ( -. N e. { N , n } -> ( V USGrph E -> ( n e. V -> -. { N , n } e. ran E ) ) ) )
14		usgraedgrnv 25116	. . . . . . . . . . . . . . . . 17 |- ( ( V USGrph E /\ { N , n } e. ran E ) -> ( N e. V /\ n e. V ) )
15	14	simpld 461	. . . . . . . . . . . . . . . 16 |- ( ( V USGrph E /\ { N , n } e. ran E ) -> N e. V )
16	15	ex 436	. . . . . . . . . . . . . . 15 |- ( V USGrph E -> ( { N , n } e. ran E -> N e. V ) )
17	16	con3d 139	. . . . . . . . . . . . . 14 |- ( V USGrph E -> ( -. N e. V -> -. { N , n } e. ran E ) )
18	17	a1i 11	. . . . . . . . . . . . 13 |- ( n e. V -> ( V USGrph E -> ( -. N e. V -> -. { N , n } e. ran E ) ) )
19	18	com13 83	. . . . . . . . . . . 12 |- ( -. N e. V -> ( V USGrph E -> ( n e. V -> -. { N , n } e. ran E ) ) )
20	19	a1d 26	. . . . . . . . . . 11 |- ( -. N e. V -> ( -. N e. { N , n } -> ( V USGrph E -> ( n e. V -> -. { N , n } e. ran E ) ) ) )
21	13, 20	pm2.61i 168	. . . . . . . . . 10 |- ( -. N e. { N , n } -> ( V USGrph E -> ( n e. V -> -. { N , n } e. ran E ) ) )
22	5, 21	sylbi 199	. . . . . . . . 9 |- ( N e/ { N , n } -> ( V USGrph E -> ( n e. V -> -. { N , n } e. ran E ) ) )
23	4, 22	syl6 34	. . . . . . . 8 |- ( { N , n } e. ran E -> ( A. x e. ran E N e/ x -> ( V USGrph E -> ( n e. V -> -. { N , n } e. ran E ) ) ) )
24	23	com13 83	. . . . . . 7 |- ( V USGrph E -> ( A. x e. ran E N e/ x -> ( { N , n } e. ran E -> ( n e. V -> -. { N , n } e. ran E ) ) ) )
25	24	imp 431	. . . . . 6 |- ( ( V USGrph E /\ A. x e. ran E N e/ x ) -> ( { N , n } e. ran E -> ( n e. V -> -. { N , n } e. ran E ) ) )
26		ax-1 6	. . . . . 6 |- ( -. { N , n } e. ran E -> ( n e. V -> -. { N , n } e. ran E ) )
27	25, 26	pm2.61d1 163	. . . . 5 |- ( ( V USGrph E /\ A. x e. ran E N e/ x ) -> ( n e. V -> -. { N , n } e. ran E ) )
28	27	ralrimiv 2802	. . . 4 |- ( ( V USGrph E /\ A. x e. ran E N e/ x ) -> A. n e. V -. { N , n } e. ran E )
29		rabeq0 3756	. . . 4 |- ( { n e. V | { N , n } e. ran E } = (/) <-> A. n e. V -. { N , n } e. ran E )
30	28, 29	sylibr 216	. . 3 |- ( ( V USGrph E /\ A. x e. ran E N e/ x ) -> { n e. V | { N , n } e. ran E } = (/) )
31	2, 30	eqtrd 2487	. 2 |- ( ( V USGrph E /\ A. x e. ran E N e/ x ) -> ( <. V , E >. Neighbors N ) = (/) )
32	31	ex 436	1 |- ( V USGrph E -> ( A. x e. ran E N e/ x -> ( <. V , E >. Neighbors N ) = (/) ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 370   /\ wa 371   = wceq 1446   e. wcel 1889   e/ wnel 2625   A.wral 2739   {crab 2743   (/)c0 3733   {cpr 3972   <.cop 3976   class class class wbr 4405   ran crn 4838  (class class class)co 6295   USGrph cusg 25069   Neighbors cnbgra 25157

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-card 8378  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-nn 10617  df-2 10675  df-n0 10877  df-z 10945  df-uz 11167  df-fz 11792  df-hash 12523  df-usgra 25072  df-nbgra 25160

This theorem is referenced by: (None)