Theorem rext 4248

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 3654	. . 3 |- x e. { x }
2		vex 2766	. . . . 5 |- x e. _V
3	2	snex 4218	. . . 4 |- { x } e. _V
4		eleq2 2260	. . . . 5 |- ( z = { x } -> ( x e. z <-> x e. { x } ) )
5		eleq2 2260	. . . . 5 |- ( z = { x } -> ( y e. z <-> y e. { x } ) )
6	4, 5	imbi12d 234	. . . 4 |- ( z = { x } -> ( ( x e. z -> y e. z ) <-> ( x e. { x } -> y e. { x } ) ) )
7	3, 6	spcv 2858	. . 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 3639	. . 3 |- ( y e. { x } <-> y = x )
10		equcomi 1718	. . 3 |- ( y = x -> x = y )
11	9, 10	sylbi 121	. 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 1362   = wceq 1364   e. wcel 2167   {csn 3622

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628

This theorem is referenced by: (None)