Theorem nprrel 5759

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

Hypotheses

Ref	Expression
nprrel12.1	⊢ 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		nprrel12.1	. . 3 ⊢ Rel 𝑅
3	2	brrelex1i 5756	. 2 ⊢ (𝐴𝑅𝐵 → 𝐴 ∈ V)
4	1, 3	mto 197	1 ⊢ ¬ 𝐴𝑅𝐵

Syntax hints:  ¬ wn 3   ∈ wcel 2108  Vcvv 3488   class class class wbr 5166  Rel wrel 5705

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-rel 5707

This theorem is referenced by: (None)