Theorem moop2 4236

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 2189	. . . 4 |- ( ( A = <. B , x >. /\ A = <. [_ y / x ]_ B , y >. ) -> <. B , x >. = <. [_ y / x ]_ B , y >. )
2		moop2.1	. . . . . 6 |- B e. _V
3		vex 2733	. . . . . 6 |- x e. _V
4	2, 3	opth 4222	. . . . 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 1443	. 2 |- A. x A. y ( ( A = <. B , x >. /\ A = <. [_ y / x ]_ B , y >. ) -> x = y )
8		nfcsb1v 3082	. . . . 5 |- F/_ x [_ y / x ]_ B
9		nfcv 2312	. . . . 5 |- F/_ x y
10	8, 9	nfop 3781	. . . 4 |- F/_ x <. [_ y / x ]_ B , y >.
11	10	nfeq2 2324	. . 3 |- F/ x A = <. [_ y / x ]_ B , y >.
12		csbeq1a 3058	. . . . 5 |- ( x = y -> B = [_ y / x ]_ B )
13		id 19	. . . . 5 |- ( x = y -> x = y )
14	12, 13	opeq12d 3773	. . . 4 |- ( x = y -> <. B , x >. = <. [_ y / x ]_ B , y >. )
15	14	eqeq2d 2182	. . 3 |- ( x = y -> ( A = <. B , x >. <-> A = <. [_ y / x ]_ B , y >. ) )
16	11, 15	mo4f 2079	. 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 1346   = wceq 1348   E*wmo 2020   e. wcel 2141   _Vcvv 2730   [_csb 3049   <.cop 3586

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592

This theorem is referenced by: (None)