Theorem moop2 1910

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(s):   x,A

Proof of Theorem moop2

Step	Hyp	Ref	Expression
1		cleqtr 1118	. . . . 5 |- ((<.B, x>. = A /\ A = <.{z | [y / x]z e. B}, y>.) -> <.B, x>. = <.{z | [y / x]z e. B}, y>.)
2		cleqcom 1103	. . . . 5 |- (A = <.B, x>. <-> <.B, x>. = A)
3	1, 2	sylanb 344	. . . 4 |- ((A = <.B, x>. /\ A = <.{z | [y / x]z e. B}, y>.) -> <.B, x>. = <.{z | [y / x]z e. B}, y>.)
4		visset 1350	. . . . 5 |- x e. V
5		visset 1350	. . . . 5 |- y e. V
6	4, 5	opth2 1909	. . . 4 |- (<.B, x>. = <.{z | [y / x]z e. B}, y>. -> x = y)
7	3, 6	syl 12	. . 3 |- ((A = <.B, x>. /\ A = <.{z | [y / x]z e. B}, y>.) -> x = y)
8	7	gen2 681	. 2 |- A.xA.y((A = <.B, x>. /\ A = <.{z | [y / x]z e. B}, y>.) -> x = y)
9		ax-17 925	. . . 4 |- (w e. A -> A.x w e. A)
10		hbs1 986	. . . . . 6 |- ([y / x]z e. B -> A.x[y / x]z e. B)
11	10	hbab 1096	. . . . 5 |- (w e. {z | [y / x]z e. B} -> A.x w e. {z | [y / x]z e. B})
12		ax-17 925	. . . . 5 |- (w e. y -> A.x w e. y)
13	11, 12	hbop 1879	. . . 4 |- (w e. <.{z | [y / x]z e. B}, y>. -> A.x w e. <.{z | [y / x]z e. B}, y>.)
14	9, 13	hbeq 1171	. . 3 |- (A = <.{z | [y / x]z e. B}, y>. -> A.x A = <.{z | [y / x]z e. B}, y>.)
15		sbab 1188	. . . . . 6 |- (x = y -> B = {z | [y / x]z e. B})
16		opeq1 1876	. . . . . 6 |- (B = {z | [y / x]z e. B} -> <.B, x>. = <.{z | [y / x]z e. B}, x>.)
17	15, 16	syl 12	. . . . 5 |- (x = y -> <.B, x>. = <.{z | [y / x]z e. B}, x>.)
18		opeq2 1877	. . . . 5 |- (x = y -> <.{z | [y / x]z e. B}, x>. = <.{z | [y / x]z e. B}, y>.)
19	17, 18	eqtrd 1128	. . . 4 |- (x = y -> <.B, x>. = <.{z | [y / x]z e. B}, y>.)
20	19	cleq2d 1112	. . 3 |- (x = y -> (A = <.B, x>. <-> A = <.{z | [y / x]z e. B}, y>.))
21	14, 20	mo4f 1028	. 2 |- (E*x A = <.B, x>. <-> A.xA.y((A = <.B, x>. /\ A = <.{z | [y / x]z e. B}, y>.) -> x = y))
22	8, 21	mpbir 165	1 |- E*x A = <.B, x>.

Syntax hints:   -> wi 2   /\ wa 196  A.wal 672   = weq 797   e. wel 803  [wsb 852  E*wmo 1008  {cab 1090   = wceq 1091   e. wcel 1092  <.cop 1810

This theorem is referenced by:  euop2 1912

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815

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