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Theorem rext 3942
 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 2554 . . . 4 x V
21snid 3394 . . 3 x {x}
3 snexgOLD 3926 . . . . 5 (x V → {x} V)
41, 3ax-mp 7 . . . 4 {x} V
5 eleq2 2098 . . . . 5 (z = {x} → (x zx {x}))
6 eleq2 2098 . . . . 5 (z = {x} → (y zy {x}))
75, 6imbi12d 223 . . . 4 (z = {x} → ((x zy z) ↔ (x {x} → y {x})))
84, 7spcv 2640 . . 3 (z(x zy z) → (x {x} → y {x}))
92, 8mpi 15 . 2 (z(x zy z) → y {x})
10 elsn 3382 . . 3 (y {x} ↔ y = x)
11 equcomi 1589 . . 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 1240   = wceq 1242   ∈ wcel 1390  Vcvv 2551  {csn 3367 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373 This theorem is referenced by: (None)
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