Theorem nbgra0nb 24728

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 24727	. . . 4 |- ( V USGrph E -> ( <. V , E >. Neighbors N ) = { n e. V | { N , n } e. ran E } )
2	1	adantr 463	. . 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 2741	. . . . . . . . . 10 |- ( x = { N , n } -> ( N e/ x <-> N e/ { N , n } ) )
4	3	rspcv 3153	. . . . . . . . 9 |- ( { N , n } e. ran E -> ( A. x e. ran E N e/ x -> N e/ { N , n } ) )
5		df-nel 2599	. . . . . . . . . 10 |- ( N e/ { N , n } <-> -. N e. { N , n } )
6		elprg 3985	. . . . . . . . . . . . . 14 |- ( N e. V -> ( N e. { N , n } <-> ( N = N \/ N = n ) ) )
7	6	notbid 292	. . . . . . . . . . . . 13 |- ( N e. V -> ( -. N e. { N , n } <-> -. ( N = N \/ N = n ) ) )
8		ioran 488	. . . . . . . . . . . . 13 |- ( -. ( N = N \/ N = n ) <-> ( -. N = N /\ -. N = n ) )
9	7, 8	syl6bb 261	. . . . . . . . . . . 12 |- ( N e. V -> ( -. N e. { N , n } <-> ( -. N = N /\ -. N = n ) ) )
10		eqid 2400	. . . . . . . . . . . . . 14 |- N = N
11	10	pm2.24i 144	. . . . . . . . . . . . 13 |- ( -. N = N -> ( -. N = n -> ( V USGrph E -> ( n e. V -> -. { N , n } e. ran E ) ) ) )
12	11	imp 427	. . . . . . . . . . . 12 |- ( ( -. N = N /\ -. N = n ) -> ( V USGrph E -> ( n e. V -> -. { N , n } e. ran E ) ) )
13	9, 12	syl6bi 228	. . . . . . . . . . 11 |- ( N e. V -> ( -. N e. { N , n } -> ( V USGrph E -> ( n e. V -> -. { N , n } e. ran E ) ) ) )
14		usgraedgrnv 24676	. . . . . . . . . . . . . . . . 17 |- ( ( V USGrph E /\ { N , n } e. ran E ) -> ( N e. V /\ n e. V ) )
15	14	simpld 457	. . . . . . . . . . . . . . . 16 |- ( ( V USGrph E /\ { N , n } e. ran E ) -> N e. V )
16	15	ex 432	. . . . . . . . . . . . . . 15 |- ( V USGrph E -> ( { N , n } e. ran E -> N e. V ) )
17	16	con3d 133	. . . . . . . . . . . . . 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 80	. . . . . . . . . . . 12 |- ( -. N e. V -> ( V USGrph E -> ( n e. V -> -. { N , n } e. ran E ) ) )
20	19	a1d 25	. . . . . . . . . . 11 |- ( -. N e. V -> ( -. N e. { N , n } -> ( V USGrph E -> ( n e. V -> -. { N , n } e. ran E ) ) ) )
21	13, 20	pm2.61i 164	. . . . . . . . . 10 |- ( -. N e. { N , n } -> ( V USGrph E -> ( n e. V -> -. { N , n } e. ran E ) ) )
22	5, 21	sylbi 195	. . . . . . . . 9 |- ( N e/ { N , n } -> ( V USGrph E -> ( n e. V -> -. { N , n } e. ran E ) ) )
23	4, 22	syl6 31	. . . . . . . 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 80	. . . . . . 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 427	. . . . . 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 159	. . . . 5 |- ( ( V USGrph E /\ A. x e. ran E N e/ x ) -> ( n e. V -> -. { N , n } e. ran E ) )
28	27	ralrimiv 2813	. . . 4 |- ( ( V USGrph E /\ A. x e. ran E N e/ x ) -> A. n e. V -. { N , n } e. ran E )
29		rabeq0 3758	. . . 4 |- ( { n e. V | { N , n } e. ran E } = (/) <-> A. n e. V -. { N , n } e. ran E )
30	28, 29	sylibr 212	. . 3 |- ( ( V USGrph E /\ A. x e. ran E N e/ x ) -> { n e. V | { N , n } e. ran E } = (/) )
31	2, 30	eqtrd 2441	. 2 |- ( ( V USGrph E /\ A. x e. ran E N e/ x ) -> ( <. V , E >. Neighbors N ) = (/) )
32	31	ex 432	1 |- ( V USGrph E -> ( A. x e. ran E N e/ x -> ( <. V , E >. Neighbors N ) = (/) ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 366   /\ wa 367   = wceq 1403   e. wcel 1840   e/ wnel 2597   A.wral 2751   {crab 2755   (/)c0 3735   {cpr 3971   <.cop 3975   class class class wbr 4392   ran crn 4941  (class class class)co 6232   USGrph cusg 24629   Neighbors cnbgra 24716

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-cnex 9496  ax-resscn 9497  ax-1cn 9498  ax-icn 9499  ax-addcl 9500  ax-addrcl 9501  ax-mulcl 9502  ax-mulrcl 9503  ax-mulcom 9504  ax-addass 9505  ax-mulass 9506  ax-distr 9507  ax-i2m1 9508  ax-1ne0 9509  ax-1rid 9510  ax-rnegex 9511  ax-rrecex 9512  ax-cnre 9513  ax-pre-lttri 9514  ax-pre-lttrn 9515  ax-pre-ltadd 9516  ax-pre-mulgt0 9517

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-nel 2599  df-ral 2756  df-rex 2757  df-reu 2758  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-int 4225  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-riota 6194  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-om 6637  df-1st 6736  df-2nd 6737  df-recs 6997  df-rdg 7031  df-1o 7085  df-er 7266  df-en 7473  df-dom 7474  df-sdom 7475  df-fin 7476  df-card 8270  df-pnf 9578  df-mnf 9579  df-xr 9580  df-ltxr 9581  df-le 9582  df-sub 9761  df-neg 9762  df-nn 10495  df-2 10553  df-n0 10755  df-z 10824  df-uz 11044  df-fz 11642  df-hash 12358  df-usgra 24632  df-nbgra 24719

This theorem is referenced by: (None)