Theorem nprrel 4738

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	|- -. A R B

Proof of Theorem nprrel

Step	Hyp	Ref	Expression
1		nprrel.2	. 2 |- -. A e. _V
2		nprrel.1	. . 3 |- Rel R
3	2	brrelex1i 4736	. 2 |- ( A R B -> A e. _V )
4	1, 3	mto 664	1 |- -. A R B

Syntax hints:   -. wn 3   e. wcel 2178   _Vcvv 2776   class class class wbr 4059   Rel wrel 4698

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-opab 4122  df-xp 4699  df-rel 4700

This theorem is referenced by: (None)