Theorem rext 3951

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		vex 2560	. . . 4 |- x e. _V
2	1	snid 3402	. . 3 |- x e. { x }
3		snexgOLD 3935	. . . . 5 |- ( x e. _V -> { x } e. _V )
4	1, 3	ax-mp 7	. . . 4 |- { x } e. _V
5		eleq2 2101	. . . . 5 |- ( z = { x } -> ( x e. z <-> x e. { x } ) )
6		eleq2 2101	. . . . 5 |- ( z = { x } -> ( y e. z <-> y e. { x } ) )
7	5, 6	imbi12d 223	. . . 4 |- ( z = { x } -> ( ( x e. z -> y e. z ) <-> ( x e. { x } -> y e. { x } ) ) )
8	4, 7	spcv 2646	. . 3 |- ( A. z ( x e. z -> y e. z ) -> ( x e. { x } -> y e. { x } ) )
9	2, 8	mpi 15	. 2 |- ( A. z ( x e. z -> y e. z ) -> y e. { x } )
10		velsn 3392	. . 3 |- ( y e. { x } <-> y = x )
11		equcomi 1592	. . 3 |- ( y = x -> x = y )
12	10, 11	sylbi 114	. 2 |- ( y e. { x } -> x = y )
13	9, 12	syl 14	1 |- ( A. z ( x e. z -> y e. z ) -> x = y )

Syntax hints:   -> wi 4   A.wal 1241   = wceq 1243   e. wcel 1393   _Vcvv 2557   {csn 3375

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381

This theorem is referenced by: (None)