Theorem opprc1 2489

Description: Expansion of an ordered pair when the first member is a proper class. See also opprc1b 2786, opprc2 2490, opprc3 2787.

Assertion

Ref	Expression
opprc1	|- (-. A e. V -> <.A, B>. = {(/), {B}})

Proof of Theorem opprc1

Step	Hyp	Ref	Expression
1		snprc 2433	. . . 4 |- (-. A e. V <-> {A} = (/))
2		preq1 2438	. . . 4 |- ({A} = (/) -> {{A}, {A, B}} = {(/), {A, B}})
3	1, 2	sylbi 199	. . 3 |- (-. A e. V -> {{A}, {A, B}} = {(/), {A, B}})
4		prprc1 2443	. . . 4 |- (-. A e. V -> {A, B} = {B})
5		preq2 2439	. . . 4 |- ({A, B} = {B} -> {(/), {A, B}} = {(/), {B}})
6	4, 5	syl 10	. . 3 |- (-. A e. V -> {(/), {A, B}} = {(/), {B}})
7	3, 6	eqtrd 1499	. 2 |- (-. A e. V -> {{A}, {A, B}} = {(/), {B}})
8		df-op 2406	. 2 |- <.A, B>. = {{A}, {A, B}}
9	7, 8	syl5eq 1511	1 |- (-. A e. V -> <.A, B>. = {(/), {B}})

Syntax hints:  -. wn 2   -> wi 3   = wceq 953   e. wcel 955  Vcvv 1802  (/)c0 2270  {csn 2399  {cpr 2400  <.cop 2401

This theorem is referenced by:  opprc1b 2786  opprc3 2787  opth2 2789

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-un 2040  df-nul 2271  df-sn 2402  df-pr 2403  df-op 2406

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