Theorem brrelex1 5708

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 5707	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
2	1	simpld 500	1 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ V)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2147  Vcvv 3454   class class class wbr 5106  Rel wrel 5660

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-xp 5661  df-rel 5662

This theorem is used by:  brrelex1i  5711  posn  5742  frsn  5744  releldm  5932  relelrn  5933  relimasn  6094  funmo  6571  ertr  8758  fsuppss  9399  dirtr  18802  eqvreltr  39814  frege129d  44953  nnfoctb  46268  clim2d  46886  climfv  46904  meadjiun  47679  caragenunicl  47737