Theorem rext 3970

Description: A theorem similar to extensionality, requiring the existence of a singleton. Exercise 8 of [TakeutiZaring] p. 16. (Contributed by NM, 10-Aug-1993.)

Assertion

Ref	Expression
rext	|- ( A. z ( x e. z -> y e. z ) -> x = y )

Distinct variable group:   x, y, z

Proof of Theorem rext

Step	Hyp	Ref	Expression
1		vsnid 3426	. . 3 |- x e. { x }
2		vex 2604	. . . . 5 |- x e. _V
3	2	snex 3957	. . . 4 |- { x } e. _V
4		eleq2 2142	. . . . 5 |- ( z = { x } -> ( x e. z <-> x e. { x } ) )
5		eleq2 2142	. . . . 5 |- ( z = { x } -> ( y e. z <-> y e. { x } ) )
6	4, 5	imbi12d 232	. . . 4 |- ( z = { x } -> ( ( x e. z -> y e. z ) <-> ( x e. { x } -> y e. { x } ) ) )
7	3, 6	spcv 2691	. . 3 |- ( A. z ( x e. z -> y e. z ) -> ( x e. { x } -> y e. { x } ) )
8	1, 7	mpi 15	. 2 |- ( A. z ( x e. z -> y e. z ) -> y e. { x } )
9		velsn 3415	. . 3 |- ( y e. { x } <-> y = x )
10		equcomi 1632	. . 3 |- ( y = x -> x = y )
11	9, 10	sylbi 119	. 2 |- ( y e. { x } -> x = y )
12	8, 11	syl 14	1 |- ( A. z ( x e. z -> y e. z ) -> x = y )

Syntax hints:   -> wi 4   A.wal 1282   = wceq 1284   e. wcel 1433   {csn 3398

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

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404

This theorem is referenced by: (None)