Theorem brprc 2661

Description: A property of proper class as the second argument of a binary relation.

Assertion

Ref	Expression
brprc	|- (-. B e. V -> (ARB <-> ARA))

Proof of Theorem brprc

Step	Hyp	Ref	Expression
1		opprc2 2499	. . 3 |- (-. B e. V -> <.A, B>. = <.A, A>.)
2	1	eleq1d 1540	. 2 |- (-. B e. V -> (<.A, B>. e. R <-> <.A, A>. e. R))
3		df-br 2620	. 2 |- (ARB <-> <.A, B>. e. R)
4		df-br 2620	. 2 |- (ARA <-> <.A, A>. e. R)
5	2, 3, 4	3bitr4g 555	1 |- (-. B e. V -> (ARB <-> ARA))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   e. wcel 958  Vcvv 1811  <.cop 2411   class class class wbr 2619

This theorem is referenced by:  vtoclrbr 3212  vtoclibr 3213  issetid 3280  f1oen2g 4394  f1domg 4396  unen 4434  sdomex 4473  numth2 4785  cardval 4826  elfzlem 6473  pstr 8652

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-nul 2281  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620

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