Theorem rext 4350

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 3737	. . 3 |- x e. { x }
2		vex 2824	. . . . 5 |- x e. _V
3	2	snex 4317	. . . 4 |- { x } e. _V
4		eleq2 2302	. . . . 5 |- ( z = { x } -> ( x e. z <-> x e. { x } ) )
5		eleq2 2302	. . . . 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 2919	. . 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 3722	. . 3 |- ( y e. { x } <-> y = x )
10		equcomi 1756	. . 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 1400   = wceq 1402   e. wcel 2209   {csn 3705

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711

This theorem is referenced by: (None)