Theorem opprc3 2792

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 2495	. . . 4 |- (-. B e. V -> <.A, B>. = <.A, A>.)
2		opprc1 2494	. . . . 5 |- (-. A e. V -> <.A, A>. = {(/), {A}})
3		snprc 2439	. . . . . 6 |- (-. A e. V <-> {A} = (/))
4		preq2 2445	. . . . . 6 |- ({A} = (/) -> {(/), {A}} = {(/), (/)})
5	3, 4	sylbi 199	. . . . 5 |- (-. A e. V -> {(/), {A}} = {(/), (/)})
6	2, 5	eqtrd 1504	. . . 4 |- (-. A e. V -> <.A, A>. = {(/), (/)})
7	1, 6	sylan9eqr 1526	. . 3 |- ((-. A e. V /\ -. B e. V) -> <.A, B>. = {(/), (/)})
8		dfsn2 2416	. . 3 |- {(/)} = {(/), (/)}
9	7, 8	syl6eqr 1522	. 2 |- ((-. A e. V /\ -. B e. V) -> <.A, B>. = {(/)})
10		0ex 2706	. . . . . 6 |- (/) e. V
11	10	snid 2431	. . . . 5 |- (/) e. {(/)}
12		eleq2 1532	. . . . 5 |- (<.A, B>. = {(/)} -> ((/) e. <.A, B>. <-> (/) e. {(/)}))
13	11, 12	mpbiri 194	. . . 4 |- (<.A, B>. = {(/)} -> (/) e. <.A, B>.)
14		opprc1b 2791	. . . 4 |- (-. A e. V <-> (/) e. <.A, B>.)
15	13, 14	sylibr 200	. . 3 |- (<.A, B>. = {(/)} -> -. A e. V)
16		opprc1 2494	. . . . . 6 |- (-. A e. V -> <.A, B>. = {(/), {B}})
17	16	eqeq1d 1480	. . . . 5 |- (-. A e. V -> (<.A, B>. = {(/)} <-> {(/), {B}} = {(/)}))
18		snex 2745	. . . . . . 7 |- {B} e. V
19	18, 10	preqr2 2478	. . . . . 6 |- ({(/), {B}} = {(/), (/)} -> {B} = (/))
20	8	eqeq2i 1482	. . . . . 6 |- ({(/), {B}} = {(/)} <-> {(/), {B}} = {(/), (/)})
21		snprc 2439	. . . . . 6 |- (-. B e. V <-> {B} = (/))
22	19, 20, 21	3imtr4 219	. . . . 5 |- ({(/), {B}} = {(/)} -> -. B e. V)
23	17, 22	syl6bi 214	. . . 4 |- (-. A e. V -> (<.A, B>. = {(/)} -> -. B e. V))
24	23	anc2li 302	. . 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 157	1 |- ((-. A e. V /\ -. B e. V) <-> <.A, B>. = {(/)})

Syntax hints:  -. wn 2   <-> wb 146   /\ wa 223   = wceq 954   e. wcel 956  Vcvv 1807  (/)c0 2276  {csn 2405  {cpr 2406  <.cop 2407

This theorem is referenced by:  dmsnsn0 3320  dmsnop 3323

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-13 967  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-sep 2698  ax-nul 2705  ax-pow 2737

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-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412

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