Theorem moop2 4006

Description: "At most one" property of an ordered pair. (Contributed by NM, 11-Apr-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)

Hypothesis

Ref	Expression
moop2.1	|- B e. _V

Assertion

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

Distinct variable group:   x, A

Allowed substitution hint:   B( x)

Proof of Theorem moop2

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqtr2 2099	. . . 4 |- ( ( A = <. B , x >. /\ A = <. [_ y / x ]_ B , y >. ) -> <. B , x >. = <. [_ y / x ]_ B , y >. )
2		moop2.1	. . . . . 6 |- B e. _V
3		vex 2604	. . . . . 6 |- x e. _V
4	2, 3	opth 3992	. . . . 5 |- ( <. B , x >. = <. [_ y / x ]_ B , y >. <-> ( B = [_ y / x ]_ B /\ x = y ) )
5	4	simprbi 269	. . . 4 |- ( <. B , x >. = <. [_ y / x ]_ B , y >. -> x = y )
6	1, 5	syl 14	. . 3 |- ( ( A = <. B , x >. /\ A = <. [_ y / x ]_ B , y >. ) -> x = y )
7	6	gen2 1379	. 2 |- A. x A. y ( ( A = <. B , x >. /\ A = <. [_ y / x ]_ B , y >. ) -> x = y )
8		nfcsb1v 2938	. . . . 5 |- F/_ x [_ y / x ]_ B
9		nfcv 2219	. . . . 5 |- F/_ x y
10	8, 9	nfop 3586	. . . 4 |- F/_ x <. [_ y / x ]_ B , y >.
11	10	nfeq2 2230	. . 3 |- F/ x A = <. [_ y / x ]_ B , y >.
12		csbeq1a 2916	. . . . 5 |- ( x = y -> B = [_ y / x ]_ B )
13		id 19	. . . . 5 |- ( x = y -> x = y )
14	12, 13	opeq12d 3578	. . . 4 |- ( x = y -> <. B , x >. = <. [_ y / x ]_ B , y >. )
15	14	eqeq2d 2092	. . 3 |- ( x = y -> ( A = <. B , x >. <-> A = <. [_ y / x ]_ B , y >. ) )
16	11, 15	mo4f 2001	. 2 |- ( E* x A = <. B , x >. <-> A. x A. y ( ( A = <. B , x >. /\ A = <. [_ y / x ]_ B , y >. ) -> x = y ) )
17	7, 16	mpbir 144	1 |- E* x A = <. B , x >.

Syntax hints:   -> wi 4   /\ wa 102   A.wal 1282   = wceq 1284   e. wcel 1433   E*wmo 1942   _Vcvv 2601   [_csb 2908   <.cop 3401

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-sbc 2816  df-csb 2909  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407

This theorem is referenced by: (None)