Theorem opprc3 2873

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 2564	. . . 4 |- (-. B e. V -> <.A, B>. = <.A, A>.)
2		opprc1 2563	. . . . 5 |- (-. A e. V -> <.A, A>. = {(/), {A}})
3		snprc 2504	. . . . . 6 |- (-. A e. V <-> {A} = (/))
4		preq2 2510	. . . . . 6 |- ({A} = (/) -> {(/), {A}} = {(/), (/)})
5	3, 4	sylbi 197	. . . . 5 |- (-. A e. V -> {(/), {A}} = {(/), (/)})
6	2, 5	eqtrd 1550	. . . 4 |- (-. A e. V -> <.A, A>. = {(/), (/)})
7	1, 6	sylan9eqr 1572	. . 3 |- ((-. A e. V /\ -. B e. V) -> <.A, B>. = {(/), (/)})
8		dfsn2 2478	. . 3 |- {(/)} = {(/), (/)}
9	7, 8	syl6eqr 1568	. 2 |- ((-. A e. V /\ -. B e. V) -> <.A, B>. = {(/)})
10		0ex 2785	. . . . . 6 |- (/) e. V
11	10	snid 2496	. . . . 5 |- (/) e. {(/)}
12		eleq2 1578	. . . . 5 |- (<.A, B>. = {(/)} -> ((/) e. <.A, B>. <-> (/) e. {(/)}))
13	11, 12	mpbiri 192	. . . 4 |- (<.A, B>. = {(/)} -> (/) e. <.A, B>.)
14		opprc1b 2872	. . . 4 |- (-. A e. V <-> (/) e. <.A, B>.)
15	13, 14	sylibr 198	. . 3 |- (<.A, B>. = {(/)} -> -. A e. V)
16		opprc1 2563	. . . . . 6 |- (-. A e. V -> <.A, B>. = {(/), {B}})
17	16	eqeq1d 1526	. . . . 5 |- (-. A e. V -> (<.A, B>. = {(/)} <-> {(/), {B}} = {(/)}))
18		snex 2826	. . . . . . 7 |- {B} e. V
19	18, 10	preqr2 2547	. . . . . 6 |- ({(/), {B}} = {(/), (/)} -> {B} = (/))
20	8	eqeq2i 1528	. . . . . 6 |- ({(/), {B}} = {(/)} <-> {(/), {B}} = {(/), (/)})
21		snprc 2504	. . . . . 6 |- (-. B e. V <-> {B} = (/))
22	19, 20, 21	3imtr4i 217	. . . . 5 |- ({(/), {B}} = {(/)} -> -. B e. V)
23	17, 22	syl6bi 212	. . . 4 |- (-. A e. V -> (<.A, B>. = {(/)} -> -. B e. V))
24	23	anc2li 300	. . 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	impbii 155	1 |- ((-. A e. V /\ -. B e. V) <-> <.A, B>. = {(/)})

Syntax hints:  -. wn 2   <-> wb 144   /\ wa 221   = wceq 992   e. wcel 994  Vcvv 1857  (/)c0 2332  {csn 2467  {cpr 2468  <.cop 2469

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-sep 2777  ax-nul 2784  ax-pow 2818

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-v 1858  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-nul 2333  df-pw 2459  df-sn 2470  df-pr 2471  df-op 2474

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