Theorem nprrel 4030

Description: No proper class is related to anything via any relation. (Contributed by Roy F. Longton, 30-Jul-2005.)

Hypotheses

Ref	Expression
nprrel.1	|- Rel R
nprrel.2	|- -. A e. _V

Assertion

Ref	Expression
nprrel	|- -. ARB

Proof of Theorem nprrel

Step	Hyp	Ref	Expression
1		nprrel.2	. 2 |- -. A e. _V
2		nprrel.1	. . 3 |- Rel R
3	2	brrelexi 4029	. 2 |- (ARB -> A e. _V)
4	1, 3	mto 121	1 |- -. ARB

Syntax hints:  -. wn 2   e. wcel 1300  _Vcvv 2292   class class class wbr 3338  Rel wrel 3991

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-xp 4000  df-rel 4001

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