Theorem opprc1 2563

Description: Expansion of an ordered pair when the first member is a proper class. See also opprc1b 2872, opprc2 2564, opprc3 2873.

Assertion

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

Proof of Theorem opprc1

Step	Hyp	Ref	Expression
1		snprc 2504	. . . 4 |- (-. A e. V <-> {A} = (/))
2		preq1 2509	. . . 4 |- ({A} = (/) -> {{A}, {A, B}} = {(/), {A, B}})
3	1, 2	sylbi 197	. . 3 |- (-. A e. V -> {{A}, {A, B}} = {(/), {A, B}})
4		prprc1 2515	. . . 4 |- (-. A e. V -> {A, B} = {B})
5		preq2 2510	. . . 4 |- ({A, B} = {B} -> {(/), {A, B}} = {(/), {B}})
6	4, 5	syl 10	. . 3 |- (-. A e. V -> {(/), {A, B}} = {(/), {B}})
7	3, 6	eqtrd 1550	. 2 |- (-. A e. V -> {{A}, {A, B}} = {(/), {B}})
8		df-op 2474	. 2 |- <.A, B>. = {{A}, {A, B}}
9	7, 8	syl5eq 1562	1 |- (-. A e. V -> <.A, B>. = {(/), {B}})

Syntax hints:  -. wn 2   -> wi 3   = wceq 992   e. wcel 994  Vcvv 1857  (/)c0 2332  {csn 2467  {cpr 2468  <.cop 2469

This theorem is referenced by:  opprc1b 2872  opprc3 2873  opth2 2876

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

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