Theorem rext 4916

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 4209	. . 3 |- x e. { x }
2		snex 4908	. . . 4 |- { x } e. _V
3		eleq2 2690	. . . . 5 |- ( z = { x } -> ( x e. z <-> x e. { x } ) )
4		eleq2 2690	. . . . 5 |- ( z = { x } -> ( y e. z <-> y e. { x } ) )
5	3, 4	imbi12d 334	. . . 4 |- ( z = { x } -> ( ( x e. z -> y e. z ) <-> ( x e. { x } -> y e. { x } ) ) )
6	2, 5	spcv 3299	. . 3 |- ( A. z ( x e. z -> y e. z ) -> ( x e. { x } -> y e. { x } ) )
7	1, 6	mpi 20	. 2 |- ( A. z ( x e. z -> y e. z ) -> y e. { x } )
8		velsn 4193	. . 3 |- ( y e. { x } <-> y = x )
9		equcomi 1944	. . 3 |- ( y = x -> x = y )
10	8, 9	sylbi 207	. 2 |- ( y e. { x } -> x = y )
11	7, 10	syl 17	1 |- ( A. z ( x e. z -> y e. z ) -> x = y )

Syntax hints:   -> wi 4   A.wal 1481   = wceq 1483   e. wcel 1990   {csn 4177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-dif 3577  df-un 3579  df-nul 3916  df-sn 4178  df-pr 4180

This theorem is referenced by: (None)