Theorem nprrel 4815

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 4813	. 2 |- ( A R B -> A e. _V )
4	1, 3	mto 672	1 |- -. A R B

Syntax hints:   -. wn 3   e. wcel 2209   _Vcvv 2821   class class class wbr 4125   Rel wrel 4774

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-xp 4775  df-rel 4776

This theorem is referenced by: (None)