Theorem opprc1 2498

Description: Expansion of an ordered pair when the first member is a proper class. See also opprc1b 2796, opprc2 2499, opprc3 2797.

Assertion

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

Proof of Theorem opprc1

Step	Hyp	Ref	Expression
1		snprc 2443	. . . 4 |- (-. A e. V <-> {A} = (/))
2		preq1 2448	. . . 4 |- ({A} = (/) -> {{A}, {A, B}} = {(/), {A, B}})
3	1, 2	sylbi 199	. . 3 |- (-. A e. V -> {{A}, {A, B}} = {(/), {A, B}})
4		prprc1 2452	. . . 4 |- (-. A e. V -> {A, B} = {B})
5		preq2 2449	. . . 4 |- ({A, B} = {B} -> {(/), {A, B}} = {(/), {B}})
6	4, 5	syl 10	. . 3 |- (-. A e. V -> {(/), {A, B}} = {(/), {B}})
7	3, 6	eqtrd 1507	. 2 |- (-. A e. V -> {{A}, {A, B}} = {(/), {B}})
8		df-op 2416	. 2 |- <.A, B>. = {{A}, {A, B}}
9	7, 8	syl5eq 1519	1 |- (-. A e. V -> <.A, B>. = {(/), {B}})

Syntax hints:  -. wn 2   -> wi 3   = wceq 956   e. wcel 958  Vcvv 1811  (/)c0 2280  {csn 2409  {cpr 2410  <.cop 2411

This theorem is referenced by:  opprc1b 2796  opprc3 2797  opth2 2800

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

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