Theorem nbgra0nb 25236

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 25235	. . . 4 |- ( V USGrph E -> ( <. V , E >. Neighbors N ) = { n e. V | { N , n } e. ran E } )
2	1	adantr 472	. . 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 2749	. . . . . . . . . 10 |- ( x = { N , n } -> ( N e/ x <-> N e/ { N , n } ) )
4	3	rspcv 3132	. . . . . . . . 9 |- ( { N , n } e. ran E -> ( A. x e. ran E N e/ x -> N e/ { N , n } ) )
5		df-nel 2644	. . . . . . . . . 10 |- ( N e/ { N , n } <-> -. N e. { N , n } )
6		elprg 3975	. . . . . . . . . . . . . 14 |- ( N e. V -> ( N e. { N , n } <-> ( N = N \/ N = n ) ) )
7	6	notbid 301	. . . . . . . . . . . . 13 |- ( N e. V -> ( -. N e. { N , n } <-> -. ( N = N \/ N = n ) ) )
8		ioran 498	. . . . . . . . . . . . 13 |- ( -. ( N = N \/ N = n ) <-> ( -. N = N /\ -. N = n ) )
9	7, 8	syl6bb 269	. . . . . . . . . . . 12 |- ( N e. V -> ( -. N e. { N , n } <-> ( -. N = N /\ -. N = n ) ) )
10		eqid 2471	. . . . . . . . . . . . . 14 |- N = N
11	10	pm2.24i 138	. . . . . . . . . . . . 13 |- ( -. N = N -> ( -. N = n -> ( V USGrph E -> ( n e. V -> -. { N , n } e. ran E ) ) ) )
12	11	imp 436	. . . . . . . . . . . 12 |- ( ( -. N = N /\ -. N = n ) -> ( V USGrph E -> ( n e. V -> -. { N , n } e. ran E ) ) )
13	9, 12	syl6bi 236	. . . . . . . . . . 11 |- ( N e. V -> ( -. N e. { N , n } -> ( V USGrph E -> ( n e. V -> -. { N , n } e. ran E ) ) ) )
14		usgraedgrnv 25183	. . . . . . . . . . . . . . . . 17 |- ( ( V USGrph E /\ { N , n } e. ran E ) -> ( N e. V /\ n e. V ) )
15	14	simpld 466	. . . . . . . . . . . . . . . 16 |- ( ( V USGrph E /\ { N , n } e. ran E ) -> N e. V )
16	15	ex 441	. . . . . . . . . . . . . . 15 |- ( V USGrph E -> ( { N , n } e. ran E -> N e. V ) )
17	16	con3d 140	. . . . . . . . . . . . . 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 82	. . . . . . . . . . . 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 169	. . . . . . . . . 10 |- ( -. N e. { N , n } -> ( V USGrph E -> ( n e. V -> -. { N , n } e. ran E ) ) )
22	5, 21	sylbi 200	. . . . . . . . 9 |- ( N e/ { N , n } -> ( V USGrph E -> ( n e. V -> -. { N , n } e. ran E ) ) )
23	4, 22	syl6 33	. . . . . . . 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 82	. . . . . . 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 436	. . . . . 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 164	. . . . 5 |- ( ( V USGrph E /\ A. x e. ran E N e/ x ) -> ( n e. V -> -. { N , n } e. ran E ) )
28	27	ralrimiv 2808	. . . 4 |- ( ( V USGrph E /\ A. x e. ran E N e/ x ) -> A. n e. V -. { N , n } e. ran E )
29		rabeq0 3757	. . . 4 |- ( { n e. V | { N , n } e. ran E } = (/) <-> A. n e. V -. { N , n } e. ran E )
30	28, 29	sylibr 217	. . 3 |- ( ( V USGrph E /\ A. x e. ran E N e/ x ) -> { n e. V | { N , n } e. ran E } = (/) )
31	2, 30	eqtrd 2505	. 2 |- ( ( V USGrph E /\ A. x e. ran E N e/ x ) -> ( <. V , E >. Neighbors N ) = (/) )
32	31	ex 441	1 |- ( V USGrph E -> ( A. x e. ran E N e/ x -> ( <. V , E >. Neighbors N ) = (/) ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 375   /\ wa 376   = wceq 1452   e. wcel 1904   e/ wnel 2642   A.wral 2756   {crab 2760   (/)c0 3722   {cpr 3961   <.cop 3965   class class class wbr 4395   ran crn 4840  (class class class)co 6308   USGrph cusg 25136   Neighbors cnbgra 25224

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-hash 12554  df-usgra 25139  df-nbgra 25227

This theorem is referenced by: (None)