Theorem opprc1b 2791

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 2494	. . 3 |- (-. A e. V -> <.A, B>. = {(/), {B}})
2		0ex 2706	. . . 4 |- (/) e. V
3	2	pri1 2446	. . 3 |- (/) e. {(/), {B}}
4	1, 3	syl5eleqr 1552	. 2 |- (-. A e. V -> (/) e. <.A, B>.)
5		opeq1 2483	. . . . . 6 |- (x = A -> <.x, B>. = <.A, B>.)
6	5	eleq2d 1538	. . . . 5 |- (x = A -> ((/) e. <.x, B>. <-> (/) e. <.A, B>.))
7	6	negbid 610	. . . 4 |- (x = A -> (-. (/) e. <.x, B>. <-> -. (/) e. <.A, B>.))
8		visset 1809	. . . . . . . . 9 |- x e. V
9	8	snnz 2454	. . . . . . . 8 |- {x} =/= (/)
10		df-ne 1584	. . . . . . . 8 |- ({x} =/= (/) <-> -. {x} = (/))
11	9, 10	mpbi 189	. . . . . . 7 |- -. {x} = (/)
12		eqcom 1474	. . . . . . 7 |- ({x} = (/) <-> (/) = {x})
13	11, 12	mtbi 191	. . . . . 6 |- -. (/) = {x}
14	8	prnz 2455	. . . . . . . 8 |- {x, B} =/= (/)
15		df-ne 1584	. . . . . . . 8 |- ({x, B} =/= (/) <-> -. {x, B} = (/))
16	14, 15	mpbi 189	. . . . . . 7 |- -. {x, B} = (/)
17		eqcom 1474	. . . . . . 7 |- ({x, B} = (/) <-> (/) = {x, B})
18	16, 17	mtbi 191	. . . . . 6 |- -. (/) = {x, B}
19	13, 18	pm3.2ni 579	. . . . 5 |- -. ((/) = {x} \/ (/) = {x, B})
20	2	elop 2778	. . . . 5 |- ((/) e. <.x, B>. <-> ((/) = {x} \/ (/) = {x, B}))
21	19, 20	mtbir 192	. . . 4 |- -. (/) e. <.x, B>.
22	7, 21	vtoclg 1843	. . 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 954   e. wcel 956   =/= wne 1582  Vcvv 1807  (/)c0 2276  {csn 2405  {cpr 2406  <.cop 2407

This theorem is referenced by:  opprc3 2792  opeqex 2793  opth2 2795  onxpdisj 3236  funopg 3539

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-nul 2705

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-un 2046  df-nul 2277  df-sn 2408  df-pr 2409  df-op 2412

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