Theorem moop2 2790

Description: "At most one" property of an ordered pair. The proof is a little tricky because we do not place any restrictions on class B.

Assertion

Ref	Expression
moop2	|- E*x A = <.B, x>.

Distinct variable group:   x,A

Proof of Theorem moop2

Step	Hyp	Ref	Expression
1		eqtr2t 1485	. . . 4 |- ((A = <.B, x>. /\ A = <.{z | [y / x]z e. B}, y>.) -> <.B, x>. = <.{z | [y / x]z e. B}, y>.)
2		visset 1804	. . . . 5 |- x e. V
3		visset 1804	. . . . 5 |- y e. V
4	2, 3	opth2 2789	. . . 4 |- (<.B, x>. = <.{z | [y / x]z e. B}, y>. -> x = y)
5	1, 4	syl 10	. . 3 |- ((A = <.B, x>. /\ A = <.{z | [y / x]z e. B}, y>.) -> x = y)
6	5	gen2 980	. 2 |- A.xA.y((A = <.B, x>. /\ A = <.{z | [y / x]z e. B}, y>.) -> x = y)
7		ax-17 968	. . . 4 |- (w e. A -> A.x w e. A)
8		hbs1 1327	. . . . . 6 |- ([y / x]z e. B -> A.x[y / x]z e. B)
9	8	hbab 1460	. . . . 5 |- (w e. {z | [y / x]z e. B} -> A.x w e. {z | [y / x]z e. B})
10		ax-17 968	. . . . 5 |- (w e. y -> A.x w e. y)
11	9, 10	hbop 2487	. . . 4 |- (w e. <.{z | [y / x]z e. B}, y>. -> A.x w e. <.{z | [y / x]z e. B}, y>.)
12	7, 11	hbeq 1557	. . 3 |- (A = <.{z | [y / x]z e. B}, y>. -> A.x A = <.{z | [y / x]z e. B}, y>.)
13		sbab 1575	. . . . . 6 |- (x = y -> B = {z | [y / x]z e. B})
14	13	opeq1d 2484	. . . . 5 |- (x = y -> <.B, x>. = <.{z | [y / x]z e. B}, x>.)
15		opeq2 2479	. . . . 5 |- (x = y -> <.{z | [y / x]z e. B}, x>. = <.{z | [y / x]z e. B}, y>.)
16	14, 15	eqtrd 1499	. . . 4 |- (x = y -> <.B, x>. = <.{z | [y / x]z e. B}, y>.)
17	16	eqeq2d 1478	. . 3 |- (x = y -> (A = <.B, x>. <-> A = <.{z | [y / x]z e. B}, y>.))
18	12, 17	mo4f 1395	. 2 |- (E*x A = <.B, x>. <-> A.xA.y((A = <.B, x>. /\ A = <.{z | [y / x]z e. B}, y>.) -> x = y))
19	6, 18	mpbir 190	1 |- E*x A = <.B, x>.

Syntax hints:   -> wi 3   /\ wa 223  A.wal 951   = wceq 953   e. wcel 955  [wsbc 1166  E*wmo 1374  {cab 1456  <.cop 2401

This theorem is referenced by:  euop2 2795

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  ax-pr 2769

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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