Theorem nbgra0nb 24093

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 24092	. . . 4 |- ( V USGrph E -> ( <. V , E >. Neighbors N ) = { n e. V | { N , n } e. ran E } )
2	1	adantr 465	. . 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 2802	. . . . . . . . . 10 |- ( x = { N , n } -> ( N e/ x <-> N e/ { N , n } ) )
4	3	rspcv 3205	. . . . . . . . 9 |- ( { N , n } e. ran E -> ( A. x e. ran E N e/ x -> N e/ { N , n } ) )
5		df-nel 2660	. . . . . . . . . 10 |- ( N e/ { N , n } <-> -. N e. { N , n } )
6		elprg 4038	. . . . . . . . . . . . . 14 |- ( N e. V -> ( N e. { N , n } <-> ( N = N \/ N = n ) ) )
7	6	notbid 294	. . . . . . . . . . . . 13 |- ( N e. V -> ( -. N e. { N , n } <-> -. ( N = N \/ N = n ) ) )
8		ioran 490	. . . . . . . . . . . . 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 2462	. . . . . . . . . . . . . 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 429	. . . . . . . . . . . 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 24041	. . . . . . . . . . . . . . . . 17 |- ( ( V USGrph E /\ { N , n } e. ran E ) -> ( N e. V /\ n e. V ) )
15	14	simpld 459	. . . . . . . . . . . . . . . 16 |- ( ( V USGrph E /\ { N , n } e. ran E ) -> N e. V )
16	15	ex 434	. . . . . . . . . . . . . . 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 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 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 429	. . . . . 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 2871	. . . 4 |- ( ( V USGrph E /\ A. x e. ran E N e/ x ) -> A. n e. V -. { N , n } e. ran E )
29		rabeq0 3802	. . . 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 2503	. 2 |- ( ( V USGrph E /\ A. x e. ran E N e/ x ) -> ( <. V , E >. Neighbors N ) = (/) )
32	31	ex 434	1 |- ( V USGrph E -> ( A. x e. ran E N e/ x -> ( <. V , E >. Neighbors N ) = (/) ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 368   /\ wa 369   = wceq 1374   e. wcel 1762   e/ wnel 2658   A.wral 2809   {crab 2813   (/)c0 3780   {cpr 4024   <.cop 4028   class class class wbr 4442   ran crn 4995  (class class class)co 6277   USGrph cusg 23995   Neighbors cnbgra 24081

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-recs 7034  df-rdg 7068  df-1o 7122  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-card 8311  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-nn 10528  df-2 10585  df-n0 10787  df-z 10856  df-uz 11074  df-fz 11664  df-hash 12363  df-usgra 23998  df-nbgra 24084

This theorem is referenced by: (None)