Theorem brrelex1 5730

Description: If two classes are related by a binary relation, then the first class is a set. (Contributed by NM, 18-May-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)

Assertion

Ref	Expression
brrelex1	⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ V)

Proof of Theorem brrelex1

Step	Hyp	Ref	Expression
1		brrelex12 5729	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
2	1	simpld 496	1 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∧ wa 397   ∈ wcel 2107  Vcvv 3475   class class class wbr 5149  Rel wrel 5682

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684

This theorem is referenced by:  brrelex1i  5733  posn  5762  frsn  5764  releldm  5944  relelrn  5945  relimasn  6084  funmo  6564  funmoOLD  6565  ertr  8718  dirtr  18555  eqvreltr  37477  fsuppss  41065  frege129d  42514  nnfoctb  43734  clim2d  44389  climfv  44407  meadjiun  45182  caragenunicl  45240