Theorem opprc2 2496

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 2449	. . . 4 |- (-. B e. V -> {B, A} = {A})
2		prcom 2444	. . . 4 |- {B, A} = {A, B}
3		dfsn2 2417	. . . 4 |- {A} = {A, A}
4	1, 2, 3	3eqtr3g 1528	. . 3 |- (-. B e. V -> {A, B} = {A, A})
5		preq2 2446	. . 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 2413	. 2 |- <.A, B>. = {{A}, {A, B}}
8		df-op 2413	. 2 |- <.A, A>. = {{A}, {A, A}}
9	6, 7, 8	3eqtr4g 1529	1 |- (-. B e. V -> <.A, B>. = <.A, A>.)

Syntax hints:  -. wn 2   -> wi 3   = wceq 955   e. wcel 957  Vcvv 1808  {csn 2406  {cpr 2407  <.cop 2408

This theorem is referenced by:  brprc 2657  opprc3 2793  relsn 3250  opelxpex2 3275  opeldm 3310  dmsnop 3324  oprprc2 3980  tpsex 7565  vcex 8163  nvex 8194

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-v 1809  df-dif 2046  df-un 2047  df-nul 2278  df-sn 2409  df-pr 2410  df-op 2413

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