Theorem rext 4200

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 3615	. . 3 |- x e. { x }
2		vex 2733	. . . . 5 |- x e. _V
3	2	snex 4171	. . . 4 |- { x } e. _V
4		eleq2 2234	. . . . 5 |- ( z = { x } -> ( x e. z <-> x e. { x } ) )
5		eleq2 2234	. . . . 5 |- ( z = { x } -> ( y e. z <-> y e. { x } ) )
6	4, 5	imbi12d 233	. . . 4 |- ( z = { x } -> ( ( x e. z -> y e. z ) <-> ( x e. { x } -> y e. { x } ) ) )
7	3, 6	spcv 2824	. . 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 3600	. . 3 |- ( y e. { x } <-> y = x )
10		equcomi 1697	. . 3 |- ( y = x -> x = y )
11	9, 10	sylbi 120	. 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 1346   = wceq 1348   e. wcel 2141   {csn 3583

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

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589

This theorem is referenced by: (None)