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Theorem unisn 3624
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 3417 . . 3 {𝐴} = {𝐴, 𝐴}
21unieqi 3618 . 2 {𝐴} = {𝐴, 𝐴}
3 unisn.1 . . 3 𝐴 ∈ V
43, 3unipr 3622 . 2 {𝐴, 𝐴} = (𝐴𝐴)
5 unidm 3114 . 2 (𝐴𝐴) = 𝐴
62, 4, 53eqtri 2080 1 {𝐴} = 𝐴
Colors of variables: wff set class
Syntax hints:   = wceq 1259  wcel 1409  Vcvv 2574  cun 2943  {csn 3403  {cpr 3404   cuni 3608
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-v 2576  df-un 2950  df-sn 3409  df-pr 3410  df-uni 3609
This theorem is referenced by:  unisng  3625  uniintsnr  3679  unisuc  4178  op1sta  4830  op2nda  4833  elxp4  4836  uniabio  4905  iotass  4912  en1bg  6311
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