Theorem opprc3 1908

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

Assertion

Ref	Expression
opprc3	|- ((-. A e. V /\ -. B e. V) <-> <.A, B>. = {(/)})

Proof of Theorem opprc3

Step	Hyp	Ref	Expression
1		opprc2 1907	. . . 4 |- (-. B e. V -> <.A, B>. = <.A, A>.)
2		opprc1 1905	. . . . 5 |- (-. A e. V -> <.A, A>. = {(/), {A}})
3		snprc 1838	. . . . . 6 |- (-. A e. V <-> {A} = (/))
4		preq2 1871	. . . . . 6 |- ({A} = (/) -> {(/), {A}} = {(/), (/)})
5	3, 4	sylbi 174	. . . . 5 |- (-. A e. V -> {(/), {A}} = {(/), (/)})
6	2, 5	eqtrd 1128	. . . 4 |- (-. A e. V -> <.A, A>. = {(/), (/)})
7	1, 6	sylan9eqr 1145	. . 3 |- ((-. A e. V /\ -. B e. V) -> <.A, B>. = {(/), (/)})
8		dfsn2 1819	. . 3 |- {(/)} = {(/), (/)}
9	7, 8	syl6eqr 1142	. 2 |- ((-. A e. V /\ -. B e. V) -> <.A, B>. = {(/)})
10		0ex 1745	. . . . . 6 |- (/) e. V
11	10	snid 1830	. . . . 5 |- (/) e. {(/)}
12		eleq2 1150	. . . . 5 |- (<.A, B>. = {(/)} -> ((/) e. <.A, B>. <-> (/) e. {(/)}))
13	11, 12	mpbiri 169	. . . 4 |- (<.A, B>. = {(/)} -> (/) e. <.A, B>.)
14		opprc1b 1906	. . . 4 |- (-. A e. V <-> (/) e. <.A, B>.)
15	13, 14	sylibr 175	. . 3 |- (<.A, B>. = {(/)} -> -. A e. V)
16		opprc1 1905	. . . . . 6 |- (-. A e. V -> <.A, B>. = {(/), {B}})
17	16	cleq1d 1109	. . . . 5 |- (-. A e. V -> (<.A, B>. = {(/)} <-> {(/), {B}} = {(/)}))
18		snex 1859	. . . . . . 7 |- {B} e. V
19	18, 10	prer2 1873	. . . . . 6 |- ({(/), {B}} = {(/), (/)} -> {B} = (/))
20	8	cleq2i 1111	. . . . . 6 |- ({(/), {B}} = {(/)} <-> {(/), {B}} = {(/), (/)})
21		snprc 1838	. . . . . 6 |- (-. B e. V <-> {B} = (/))
22	19, 20, 21	3imtr4 192	. . . . 5 |- ({(/), {B}} = {(/)} -> -. B e. V)
23	17, 22	syl6bi 187	. . . 4 |- (-. A e. V -> (<.A, B>. = {(/)} -> -. B e. V))
24	23	anc2li 250	. . 3 |- (-. A e. V -> (<.A, B>. = {(/)} -> (-. A e. V /\ -. B e. V)))
25	15, 24	mpcom 49	. 2 |- (<.A, B>. = {(/)} -> (-. A e. V /\ -. B e. V))
26	9, 25	impbi 139	1 |- ((-. A e. V /\ -. B e. V) <-> <.A, B>. = {(/)})

Syntax hints:  -. wn 1   <-> wb 127   /\ wa 196   = wceq 1091   e. wcel 1092  Vcvv 1348  (/)c0 1707  {csn 1808  {cpr 1809  <.cop 1810

This theorem is referenced by:  dmsnsn0 2544  dmsnop 2547

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815

metamath.org