Theorem opprc3 2803

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 2503	. . . 4 |- (-. B e. V -> <.A, B>. = <.A, A>.)
2		opprc1 2502	. . . . 5 |- (-. A e. V -> <.A, A>. = {(/), {A}})
3		snprc 2447	. . . . . 6 |- (-. A e. V <-> {A} = (/))
4		preq2 2453	. . . . . 6 |- ({A} = (/) -> {(/), {A}} = {(/), (/)})
5	3, 4	sylbi 199	. . . . 5 |- (-. A e. V -> {(/), {A}} = {(/), (/)})
6	2, 5	eqtrd 1510	. . . 4 |- (-. A e. V -> <.A, A>. = {(/), (/)})
7	1, 6	sylan9eqr 1532	. . 3 |- ((-. A e. V /\ -. B e. V) -> <.A, B>. = {(/), (/)})
8		dfsn2 2424	. . 3 |- {(/)} = {(/), (/)}
9	7, 8	syl6eqr 1528	. 2 |- ((-. A e. V /\ -. B e. V) -> <.A, B>. = {(/)})
10		0ex 2716	. . . . . 6 |- (/) e. V
11	10	snid 2439	. . . . 5 |- (/) e. {(/)}
12		eleq2 1538	. . . . 5 |- (<.A, B>. = {(/)} -> ((/) e. <.A, B>. <-> (/) e. {(/)}))
13	11, 12	mpbiri 194	. . . 4 |- (<.A, B>. = {(/)} -> (/) e. <.A, B>.)
14		opprc1b 2802	. . . 4 |- (-. A e. V <-> (/) e. <.A, B>.)
15	13, 14	sylibr 200	. . 3 |- (<.A, B>. = {(/)} -> -. A e. V)
16		opprc1 2502	. . . . . 6 |- (-. A e. V -> <.A, B>. = {(/), {B}})
17	16	eqeq1d 1486	. . . . 5 |- (-. A e. V -> (<.A, B>. = {(/)} <-> {(/), {B}} = {(/)}))
18		snex 2756	. . . . . . 7 |- {B} e. V
19	18, 10	preqr2 2486	. . . . . 6 |- ({(/), {B}} = {(/), (/)} -> {B} = (/))
20	8	eqeq2i 1488	. . . . . 6 |- ({(/), {B}} = {(/)} <-> {(/), {B}} = {(/), (/)})
21		snprc 2447	. . . . . 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 958   e. wcel 960  Vcvv 1814  (/)c0 2283  {csn 2413  {cpr 2414  <.cop 2415

This theorem is referenced by:  dmsnsn0 3331  dmsnop 3334

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-nul 2715  ax-pow 2748

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420

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