Theorem brrelex1 5713

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 5712	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
2	1	simpld 499	1 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ V)

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   ∈ wcel 2142  Vcvv 3454   class class class wbr 5108  Rel wrel 5665

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5256  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-xp 5666  df-rel 5667

This theorem is used by:  brrelex1i  5716  posn  5746  frsn  5748  releldm  5933  relelrn  5934  relimasn  6086  funmo  6552  ertr  8708  fsuppss  9341  dirtr  18664  eqvreltr  39368  frege129d  44517  nnfoctb  45796  clim2d  46415  climfv  46433  meadjiun  47208  caragenunicl  47266