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Theorem unisn 3587
Description: A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53. (Contributed by NM, 30-Aug-1993.)
Hypothesis
Ref Expression
unisn.1 A V
Assertion
Ref Expression
unisn {A} = A

Proof of Theorem unisn
StepHypRef Expression
1 dfsn2 3381 . . 3 {A} = {A, A}
21unieqi 3581 . 2 {A} = {A, A}
3 unisn.1 . . 3 A V
43, 3unipr 3585 . 2 {A, A} = (AA)
5 unidm 3080 . 2 (AA) = A
62, 4, 53eqtri 2061 1 {A} = A
Colors of variables: wff set class
Syntax hints:   = wceq 1242   wcel 1390  Vcvv 2551  cun 2909  {csn 3367  {cpr 3368   cuni 3571
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-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
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-rex 2306  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374  df-uni 3572
This theorem is referenced by:  unisng  3588  uniintsnr  3642  unisuc  4116  op1sta  4745  op2nda  4748  elxp4  4751  uniabio  4820  iotass  4827  en1bg  6216
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