Theorem moop2 2807

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 1496	. . . 4 |- ((A = <.B, x>. /\ A = <.{z | [y / x]z e. B}, y>.) -> <.B, x>. = <.{z | [y / x]z e. B}, y>.)
2		visset 1816	. . . . 5 |- x e. V
3		visset 1816	. . . . 5 |- y e. V
4	2, 3	opth2 2806	. . . 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 985	. 2 |- A.xA.y((A = <.B, x>. /\ A = <.{z | [y / x]z e. B}, y>.) -> x = y)
7		ax-17 973	. . . 4 |- (w e. A -> A.x w e. A)
8		hbs1 1334	. . . . . 6 |- ([y / x]z e. B -> A.x[y / x]z e. B)
9	8	hbab 1470	. . . . 5 |- (w e. {z | [y / x]z e. B} -> A.x w e. {z | [y / x]z e. B})
10		ax-17 973	. . . . 5 |- (w e. y -> A.x w e. y)
11	9, 10	hbop 2500	. . . 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 1568	. . 3 |- (A = <.{z | [y / x]z e. B}, y>. -> A.x A = <.{z | [y / x]z e. B}, y>.)
13		sbab 1586	. . . . . 6 |- (x = y -> B = {z | [y / x]z e. B})
14	13	opeq1d 2497	. . . . 5 |- (x = y -> <.B, x>. = <.{z | [y / x]z e. B}, x>.)
15		opeq2 2492	. . . . 5 |- (x = y -> <.{z | [y / x]z e. B}, x>. = <.{z | [y / x]z e. B}, y>.)
16	14, 15	eqtrd 1510	. . . 4 |- (x = y -> <.B, x>. = <.{z | [y / x]z e. B}, y>.)
17	16	eqeq2d 1489	. . 3 |- (x = y -> (A = <.B, x>. <-> A = <.{z | [y / x]z e. B}, y>.))
18	12, 17	mo4f 1404	. 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 956   = wceq 958   e. wcel 960  [wsbc 1172  E*wmo 1383  {cab 1466  <.cop 2415

This theorem is referenced by:  euop2 2812

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  ax-pr 2785

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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