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Theorem rext 4238
 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
Distinct variable group:   ,,

Proof of Theorem rext
StepHypRef Expression
1 vex 2804 . . . 4
21snid 3680 . . 3
3 snex 4232 . . . 4
4 eleq2 2357 . . . . 5
5 eleq2 2357 . . . . 5
64, 5imbi12d 311 . . . 4
73, 6spcv 2887 . . 3
82, 7mpi 16 . 2
9 elsn 3668 . . 3
10 equcomi 1664 . . 3
119, 10sylbi 187 . 2
128, 11syl 15 1
 Colors of variables: wff set class Syntax hints:   wi 4  wal 1530   wceq 1632   wcel 1696  csn 3653 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-v 2803  df-dif 3168  df-un 3170  df-nul 3469  df-sn 3659  df-pr 3660
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