Theorem nprrel 4542

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 ∈ V

Assertion

Ref	Expression
nprrel	⊢ ¬ 𝐴𝑅𝐵

Proof of Theorem nprrel

Step	Hyp	Ref	Expression
1		nprrel.2	. 2 ⊢ ¬ 𝐴 ∈ V
2		nprrel.1	. . 3 ⊢ Rel 𝑅
3	2	brrelex1i 4540	. 2 ⊢ (𝐴𝑅𝐵 → 𝐴 ∈ V)
4	1, 3	mto 634	1 ⊢ ¬ 𝐴𝑅𝐵

Syntax hints:  ¬ wn 3   ∈ wcel 1461  Vcvv 2655   class class class wbr 3893  Rel wrel 4502

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089

This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-v 2657  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-br 3894  df-opab 3948  df-xp 4503  df-rel 4504

This theorem is referenced by: (None)