Theorem moop2 2801

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 1493	. . . 4 |- ((A = <.B, x>. /\ A = <.{z | [y / x]z e. B}, y>.) -> <.B, x>. = <.{z | [y / x]z e. B}, y>.)
2		visset 1813	. . . . 5 |- x e. V
3		visset 1813	. . . . 5 |- y e. V
4	2, 3	opth2 2800	. . . 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 983	. 2 |- A.xA.y((A = <.B, x>. /\ A = <.{z | [y / x]z e. B}, y>.) -> x = y)
7		ax-17 971	. . . 4 |- (w e. A -> A.x w e. A)
8		hbs1 1332	. . . . . 6 |- ([y / x]z e. B -> A.x[y / x]z e. B)
9	8	hbab 1467	. . . . 5 |- (w e. {z | [y / x]z e. B} -> A.x w e. {z | [y / x]z e. B})
10		ax-17 971	. . . . 5 |- (w e. y -> A.x w e. y)
11	9, 10	hbop 2496	. . . 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 1565	. . 3 |- (A = <.{z | [y / x]z e. B}, y>. -> A.x A = <.{z | [y / x]z e. B}, y>.)
13		sbab 1583	. . . . . 6 |- (x = y -> B = {z | [y / x]z e. B})
14	13	opeq1d 2493	. . . . 5 |- (x = y -> <.B, x>. = <.{z | [y / x]z e. B}, x>.)
15		opeq2 2488	. . . . 5 |- (x = y -> <.{z | [y / x]z e. B}, x>. = <.{z | [y / x]z e. B}, y>.)
16	14, 15	eqtrd 1507	. . . 4 |- (x = y -> <.B, x>. = <.{z | [y / x]z e. B}, y>.)
17	16	eqeq2d 1486	. . 3 |- (x = y -> (A = <.B, x>. <-> A = <.{z | [y / x]z e. B}, y>.))
18	12, 17	mo4f 1402	. 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 954   = wceq 956   e. wcel 958  [wsbc 1170  E*wmo 1381  {cab 1463  <.cop 2411

This theorem is referenced by:  euop2 2806

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416

Copyright terms: Public domain