Theorem opprc1b 2786

Description: A property of an ordered pair of proper classes (due to our particular definition of ordered pair).

Assertion

Ref	Expression
opprc1b	|- (-. A e. V <-> (/) e. <.A, B>.)

Proof of Theorem opprc1b

Step	Hyp	Ref	Expression
1		opprc1 2489	. . 3 |- (-. A e. V -> <.A, B>. = {(/), {B}})
2		0ex 2701	. . . 4 |- (/) e. V
3	2	pri1 2441	. . 3 |- (/) e. {(/), {B}}
4	1, 3	syl5eleqr 1547	. 2 |- (-. A e. V -> (/) e. <.A, B>.)
5		opeq1 2478	. . . . . 6 |- (x = A -> <.x, B>. = <.A, B>.)
6	5	eleq2d 1533	. . . . 5 |- (x = A -> ((/) e. <.x, B>. <-> (/) e. <.A, B>.))
7	6	negbid 609	. . . 4 |- (x = A -> (-. (/) e. <.x, B>. <-> -. (/) e. <.A, B>.))
8		visset 1804	. . . . . . . . 9 |- x e. V
9	8	snnz 2449	. . . . . . . 8 |- {x} =/= (/)
10		df-ne 1579	. . . . . . . 8 |- ({x} =/= (/) <-> -. {x} = (/))
11	9, 10	mpbi 189	. . . . . . 7 |- -. {x} = (/)
12		eqcom 1469	. . . . . . 7 |- ({x} = (/) <-> (/) = {x})
13	11, 12	mtbi 191	. . . . . 6 |- -. (/) = {x}
14	8	prnz 2450	. . . . . . . 8 |- {x, B} =/= (/)
15		df-ne 1579	. . . . . . . 8 |- ({x, B} =/= (/) <-> -. {x, B} = (/))
16	14, 15	mpbi 189	. . . . . . 7 |- -. {x, B} = (/)
17		eqcom 1469	. . . . . . 7 |- ({x, B} = (/) <-> (/) = {x, B})
18	16, 17	mtbi 191	. . . . . 6 |- -. (/) = {x, B}
19	13, 18	pm3.2ni 578	. . . . 5 |- -. ((/) = {x} \/ (/) = {x, B})
20	2	elop 2773	. . . . 5 |- ((/) e. <.x, B>. <-> ((/) = {x} \/ (/) = {x, B}))
21	19, 20	mtbir 192	. . . 4 |- -. (/) e. <.x, B>.
22	7, 21	vtoclg 1838	. . 3 |- (A e. V -> -. (/) e. <.A, B>.)
23	22	con2i 97	. 2 |- ((/) e. <.A, B>. -> -. A e. V)
24	4, 23	impbi 157	1 |- (-. A e. V <-> (/) e. <.A, B>.)

Syntax hints:  -. wn 2   <-> wb 146   \/ wo 222   = wceq 953   e. wcel 955   =/= wne 1577  Vcvv 1802  (/)c0 2270  {csn 2399  {cpr 2400  <.cop 2401

This theorem is referenced by:  opprc3 2787  opth2 2789  onxpdisj 3231  funopg 3533

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-11 964  ax-12 965  ax-14 967  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  ax-nul 2700

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  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

Copyright terms: Public domain