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Theorem opprc3 2787
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
StepHypRef Expression
1 opprc2 2490 . . . 4 |- (-. B e. V -> <.A, B>. = <.A, A>.)
2 opprc1 2489 . . . . 5 |- (-. A e. V -> <.A, A>. = {(/), {A}})
3 snprc 2433 . . . . . 6 |- (-. A e. V <-> {A} = (/))
4 preq2 2439 . . . . . 6 |- ({A} = (/) -> {(/), {A}} = {(/), (/)})
53, 4sylbi 199 . . . . 5 |- (-. A e. V -> {(/), {A}} = {(/), (/)})
62, 5eqtrd 1499 . . . 4 |- (-. A e. V -> <.A, A>. = {(/), (/)})
71, 6sylan9eqr 1521 . . 3 |- ((-. A e. V /\ -. B e. V) -> <.A, B>. = {(/), (/)})
8 dfsn2 2410 . . 3 |- {(/)} = {(/), (/)}
97, 8syl6eqr 1517 . 2 |- ((-. A e. V /\ -. B e. V) -> <.A, B>. = {(/)})
10 0ex 2701 . . . . . 6 |- (/) e. V
1110snid 2425 . . . . 5 |- (/) e. {(/)}
12 eleq2 1527 . . . . 5 |- (<.A, B>. = {(/)} -> ((/) e. <.A, B>. <-> (/) e. {(/)}))
1311, 12mpbiri 194 . . . 4 |- (<.A, B>. = {(/)} -> (/) e. <.A, B>.)
14 opprc1b 2786 . . . 4 |- (-. A e. V <-> (/) e. <.A, B>.)
1513, 14sylibr 200 . . 3 |- (<.A, B>. = {(/)} -> -. A e. V)
16 opprc1 2489 . . . . . 6 |- (-. A e. V -> <.A, B>. = {(/), {B}})
1716eqeq1d 1475 . . . . 5 |- (-. A e. V -> (<.A, B>. = {(/)} <-> {(/), {B}} = {(/)}))
18 snex 2740 . . . . . . 7 |- {B} e. V
1918, 10preqr2 2473 . . . . . 6 |- ({(/), {B}} = {(/), (/)} -> {B} = (/))
208eqeq2i 1477 . . . . . 6 |- ({(/), {B}} = {(/)} <-> {(/), {B}} = {(/), (/)})
21 snprc 2433 . . . . . 6 |- (-. B e. V <-> {B} = (/))
2219, 20, 213imtr4 219 . . . . 5 |- ({(/), {B}} = {(/)} -> -. B e. V)
2317, 22syl6bi 214 . . . 4 |- (-. A e. V -> (<.A, B>. = {(/)} -> -. B e. V))
2423anc2li 302 . . 3 |- (-. A e. V -> (<.A, B>. = {(/)} -> (-. A e. V /\ -. B e. V)))
2515, 24mpcom 49 . 2 |- (<.A, B>. = {(/)} -> (-. A e. V /\ -. B e. V))
269, 25impbi 157 1 |- ((-. A e. V /\ -. B e. V) <-> <.A, B>. = {(/)})
Colors of variables: wff set class
Syntax hints:  -. wn 2   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  Vcvv 1802  (/)c0 2270  {csn 2399  {cpr 2400  <.cop 2401
This theorem is referenced by:  dmsnsn0 3314  dmsnop 3317
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-13 966  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-sep 2693  ax-nul 2700  ax-pow 2732
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-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406
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