Theorem opprc1 3170

Description: Expansion of an ordered pair when the first member is a proper class. See also opprc1b 3542, opprc2 3171, opprc3 3543.

Assertion

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

Proof of Theorem opprc1

Step	Hyp	Ref	Expression
1		snprc 3092	. . . 4 |- (-. A e. _V <-> {A} = (/))
2		preq1 3098	. . . 4 |- ({A} = (/) -> {{A}, {A, B}} = {(/), {A, B}})
3	1, 2	sylbi 216	. . 3 |- (-. A e. _V -> {{A}, {A, B}} = {(/), {A, B}})
4		prprc1 3108	. . . 4 |- (-. A e. _V -> {A, B} = {B})
5		preq2 3099	. . . 4 |- ({A, B} = {B} -> {(/), {A, B}} = {(/), {B}})
6	4, 5	syl 12	. . 3 |- (-. A e. _V -> {(/), {A, B}} = {(/), {B}})
7	3, 6	eqtrd 1925	. 2 |- (-. A e. _V -> {{A}, {A, B}} = {(/), {B}})
8		df-op 3053	. 2 |- <.A, B>. = {{A}, {A, B}}
9	7, 8	syl5eq 1940	1 |- (-. A e. _V -> <.A, B>. = {(/), {B}})

Syntax hints:  -. wn 2   -> wi 3   = wceq 1298   e. wcel 1300  _Vcvv 2292  (/)c0 2875  {csn 3044  {cpr 3045  <.cop 3046

This theorem is referenced by:  opprc1b 3542  opprc3 3543  opth2 3546

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-dif 2597  df-un 2600  df-nul 2876  df-sn 3049  df-pr 3050  df-op 3053

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