Theorem moop2 3548

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		eqtr2 1905	. . . 4 |- ((A = <.B, x>. /\ A = <.[_y / x]_B, y>.) -> <.B, x>. = <.[_y / x]_B, y>.)
2		visset 2295	. . . . 5 |- x e. _V
3		visset 2295	. . . . 5 |- y e. _V
4	2, 3	opth2 3546	. . . 4 |- (<.B, x>. = <.[_y / x]_B, y>. -> x = y)
5	1, 4	syl 12	. . 3 |- ((A = <.B, x>. /\ A = <.[_y / x]_B, y>.) -> x = y)
6	5	gen2 1329	. 2 |- A.xA.y((A = <.B, x>. /\ A = <.[_y / x]_B, y>.) -> x = y)
7		ax-17 1317	. . . 4 |- (z e. A -> A.x z e. A)
8		ax-17 1317	. . . . . 6 |- (z e. y -> A.x z e. y)
9	3, 8	hbcsb1 2568	. . . . 5 |- (z e. [_y / x]_B -> A.x z e. [_y / x]_B)
10	9, 8	hbop 3168	. . . 4 |- (z e. <.[_y / x]_B, y>. -> A.x z e. <.[_y / x]_B, y>.)
11	7, 10	hbeq 1995	. . 3 |- (A = <.[_y / x]_B, y>. -> A.x A = <.[_y / x]_B, y>.)
12		csbeq1a 2546	. . . . . 6 |- (x = y -> B = [_y / x]_B)
13	12	opeq1d 3164	. . . . 5 |- (x = y -> <.B, x>. = <.[_y / x]_B, x>.)
14		opeq2 3159	. . . . 5 |- (x = y -> <.[_y / x]_B, x>. = <.[_y / x]_B, y>.)
15	13, 14	eqtrd 1925	. . . 4 |- (x = y -> <.B, x>. = <.[_y / x]_B, y>.)
16	15	eqeq2d 1895	. . 3 |- (x = y -> (A = <.B, x>. <-> A = <.[_y / x]_B, y>.))
17	11, 16	mo4f 1798	. 2 |- (E*x A = <.B, x>. <-> A.xA.y((A = <.B, x>. /\ A = <.[_y / x]_B, y>.) -> x = y))
18	6, 17	mpbir 207	1 |- E*x A = <.B, x>.

Syntax hints:   -> wi 3   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  E*wmo 1772  [_csb 2540  <.cop 3046

This theorem is referenced by:  euop2 3553

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053

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