Theorem moop2 4211

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 2176	. . . 4 |- ( ( A = <. B , x >. /\ A = <. [_ y / x ]_ B , y >. ) -> <. B , x >. = <. [_ y / x ]_ B , y >. )
2		moop2.1	. . . . . 6 |- B e. _V
3		vex 2715	. . . . . 6 |- x e. _V
4	2, 3	opth 4197	. . . . 5 |- ( <. B , x >. = <. [_ y / x ]_ B , y >. <-> ( B = [_ y / x ]_ B /\ x = y ) )
5	4	simprbi 273	. . . 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 1430	. 2 |- A. x A. y ( ( A = <. B , x >. /\ A = <. [_ y / x ]_ B , y >. ) -> x = y )
8		nfcsb1v 3064	. . . . 5 |- F/_ x [_ y / x ]_ B
9		nfcv 2299	. . . . 5 |- F/_ x y
10	8, 9	nfop 3757	. . . 4 |- F/_ x <. [_ y / x ]_ B , y >.
11	10	nfeq2 2311	. . 3 |- F/ x A = <. [_ y / x ]_ B , y >.
12		csbeq1a 3040	. . . . 5 |- ( x = y -> B = [_ y / x ]_ B )
13		id 19	. . . . 5 |- ( x = y -> x = y )
14	12, 13	opeq12d 3749	. . . 4 |- ( x = y -> <. B , x >. = <. [_ y / x ]_ B , y >. )
15	14	eqeq2d 2169	. . 3 |- ( x = y -> ( A = <. B , x >. <-> A = <. [_ y / x ]_ B , y >. ) )
16	11, 15	mo4f 2066	. 2 |- ( E* x A = <. B , x >. <-> A. x A. y ( ( A = <. B , x >. /\ A = <. [_ y / x ]_ B , y >. ) -> x = y ) )
17	7, 16	mpbir 145	1 |- E* x A = <. B , x >.

Syntax hints:   -> wi 4   /\ wa 103   A.wal 1333   = wceq 1335   E*wmo 2007   e. wcel 2128   _Vcvv 2712   [_csb 3031   <.cop 3563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-sbc 2938  df-csb 3032  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569

This theorem is referenced by: (None)