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Theorem unisng 3571
Description: A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53. (Contributed by NM, 13-Aug-2002.)
Assertion
Ref Expression
unisng (A 𝑉 {A} = A)

Proof of Theorem unisng
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 sneq 3361 . . . 4 (x = A → {x} = {A})
21unieqd 3565 . . 3 (x = A {x} = {A})
3 id 19 . . 3 (x = Ax = A)
42, 3eqeq12d 2036 . 2 (x = A → ( {x} = x {A} = A))
5 vex 2538 . . 3 x V
65unisn 3570 . 2 {x} = x
74, 6vtoclg 2590 1 (A 𝑉 {A} = A)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1228   wcel 1374  {csn 3350   cuni 3554
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-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
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-rex 2290  df-v 2537  df-un 2899  df-sn 3356  df-pr 3357  df-uni 3555
This theorem is referenced by:  dfnfc2  3572  unisucg  4100  unisn3  4130  opswapg  4734  funfvdm  5161
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