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Theorem unisn 3593
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  e. 
_V
Assertion
Ref Expression
unisn  |-  U. { A }  =  A

Proof of Theorem unisn
StepHypRef Expression
1 dfsn2 3386 . . 3  |-  { A }  =  { A ,  A }
21unieqi 3587 . 2  |-  U. { A }  =  U. { A ,  A }
3 unisn.1 . . 3  |-  A  e. 
_V
43, 3unipr 3591 . 2  |-  U. { A ,  A }  =  ( A  u.  A )
5 unidm 3083 . 2  |-  ( A  u.  A )  =  A
62, 4, 53eqtri 2064 1  |-  U. { A }  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1243    e. wcel 1393   _Vcvv 2554    u. cun 2912   {csn 3372   {cpr 3373   U.cuni 3577
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2309  df-v 2556  df-un 2919  df-sn 3378  df-pr 3379  df-uni 3578
This theorem is referenced by:  unisng  3594  uniintsnr  3648  unisuc  4146  op1sta  4789  op2nda  4792  elxp4  4795  uniabio  4864  iotass  4871  en1bg  6267
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