Theorem brrelex12i 4735

Description: Two classes that are related by a binary relation are sets. (An artifact of our ordered pair definition.) (Contributed by BJ, 3-Oct-2022.)

Hypothesis

Ref	Expression
brrelexi.1	|- Rel R

Assertion

Ref	Expression
brrelex12i	|- ( A R B -> ( A e. _V /\ B e. _V ) )

Proof of Theorem brrelex12i

Step	Hyp	Ref	Expression
1		brrelexi.1	. 2 |- Rel R
2		brrelex12 4731	. 2 |- ( ( Rel R /\ A R B ) -> ( A e. _V /\ B e. _V ) )
3	1, 2	mpan 424	1 |- ( A R B -> ( A e. _V /\ B e. _V ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2178   _Vcvv 2776   class class class wbr 4059   Rel wrel 4698

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-opab 4122  df-xp 4699  df-rel 4700

This theorem is referenced by:  brabv  4823  isstruct2im  12957