Theorem rext 4209

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 3621	. . 3 |- x e. { x }
2		vex 2738	. . . . 5 |- x e. _V
3	2	snex 4180	. . . 4 |- { x } e. _V
4		eleq2 2239	. . . . 5 |- ( z = { x } -> ( x e. z <-> x e. { x } ) )
5		eleq2 2239	. . . . 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 2829	. . 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 3606	. . 3 |- ( y e. { x } <-> y = x )
10		equcomi 1702	. . 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 1351   = wceq 1353   e. wcel 2146   {csn 3589

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-v 2737  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595

This theorem is referenced by: (None)