Theorem opprc2 2564

Description: A property of an ordered pair of proper classes (due to our particular definition of ordered pair).

Assertion

Ref	Expression
opprc2	|- (-. B e. V -> <.A, B>. = <.A, A>.)

Proof of Theorem opprc2

Step	Hyp	Ref	Expression
1		prprc1 2515	. . . 4 |- (-. B e. V -> {B, A} = {A})
2		prcom 2508	. . . 4 |- {B, A} = {A, B}
3		dfsn2 2478	. . . 4 |- {A} = {A, A}
4	1, 2, 3	3eqtr3g 1573	. . 3 |- (-. B e. V -> {A, B} = {A, A})
5		preq2 2510	. . 3 |- ({A, B} = {A, A} -> {{A}, {A, B}} = {{A}, {A, A}})
6	4, 5	syl 10	. 2 |- (-. B e. V -> {{A}, {A, B}} = {{A}, {A, A}})
7		df-op 2474	. 2 |- <.A, B>. = {{A}, {A, B}}
8		df-op 2474	. 2 |- <.A, A>. = {{A}, {A, A}}
9	6, 7, 8	3eqtr4g 1574	1 |- (-. B e. V -> <.A, B>. = <.A, A>.)

Syntax hints:  -. wn 2   -> wi 3   = wceq 992   e. wcel 994  Vcvv 1857  {csn 2467  {cpr 2468  <.cop 2469

This theorem is referenced by:  brprc 2734  opprc3 2873  relsn 3343  opelxpex2 3369  opeldm 3405  dmsnop 3577  oprprc2 4043  tpsex 7817  vcex 8446  nvex 8477

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-10 1002  ax-12 1004  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-ex 1017  df-sb 1209  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-v 1858  df-dif 2101  df-un 2102  df-nul 2333  df-sn 2470  df-pr 2471  df-op 2474

Copyright terms: Public domain