Theorem moop2 4309

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 2225	. . . 4 |- ( ( A = <. B , x >. /\ A = <. [_ y / x ]_ B , y >. ) -> <. B , x >. = <. [_ y / x ]_ B , y >. )
2		moop2.1	. . . . . 6 |- B e. _V
3		vex 2776	. . . . . 6 |- x e. _V
4	2, 3	opth 4294	. . . . 5 |- ( <. B , x >. = <. [_ y / x ]_ B , y >. <-> ( B = [_ y / x ]_ B /\ x = y ) )
5	4	simprbi 275	. . . 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 1474	. 2 |- A. x A. y ( ( A = <. B , x >. /\ A = <. [_ y / x ]_ B , y >. ) -> x = y )
8		nfcsb1v 3130	. . . . 5 |- F/_ x [_ y / x ]_ B
9		nfcv 2349	. . . . 5 |- F/_ x y
10	8, 9	nfop 3844	. . . 4 |- F/_ x <. [_ y / x ]_ B , y >.
11	10	nfeq2 2361	. . 3 |- F/ x A = <. [_ y / x ]_ B , y >.
12		csbeq1a 3106	. . . . 5 |- ( x = y -> B = [_ y / x ]_ B )
13		id 19	. . . . 5 |- ( x = y -> x = y )
14	12, 13	opeq12d 3836	. . . 4 |- ( x = y -> <. B , x >. = <. [_ y / x ]_ B , y >. )
15	14	eqeq2d 2218	. . 3 |- ( x = y -> ( A = <. B , x >. <-> A = <. [_ y / x ]_ B , y >. ) )
16	11, 15	mo4f 2115	. 2 |- ( E* x A = <. B , x >. <-> A. x A. y ( ( A = <. B , x >. /\ A = <. [_ y / x ]_ B , y >. ) -> x = y ) )
17	7, 16	mpbir 146	1 |- E* x A = <. B , x >.

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1371   = wceq 1373   E*wmo 2056   e. wcel 2177   _Vcvv 2773   [_csb 3097   <.cop 3641

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4173  ax-pow 4229  ax-pr 4264

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-sbc 3003  df-csb 3098  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647

This theorem is referenced by: (None)