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Theorem rext 3925
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 (z(x zy z) → x = y)
Distinct variable group:   x,y,z

Proof of Theorem rext
StepHypRef Expression
1 vex 2538 . . . 4 x V
21snid 3377 . . 3 x {x}
3 snexgOLD 3909 . . . . 5 (x V → {x} V)
41, 3ax-mp 7 . . . 4 {x} V
5 eleq2 2083 . . . . 5 (z = {x} → (x zx {x}))
6 eleq2 2083 . . . . 5 (z = {x} → (y zy {x}))
75, 6imbi12d 223 . . . 4 (z = {x} → ((x zy z) ↔ (x {x} → y {x})))
84, 7spcv 2623 . . 3 (z(x zy z) → (x {x} → y {x}))
92, 8mpi 15 . 2 (z(x zy z) → y {x})
10 elsn 3365 . . 3 (y {x} ↔ y = x)
11 equcomi 1574 . . 3 (y = xx = y)
1210, 11sylbi 114 . 2 (y {x} → x = y)
139, 12syl 14 1 (z(x zy z) → x = y)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1226   = wceq 1228   wcel 1374  Vcvv 2535  {csn 3350
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356
This theorem is referenced by: (None)
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