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Theorem unisng 3908
 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 3744 . . . 4 (x = A → {x} = {A})
21unieqd 3902 . . 3 (x = A{x} = {A})
3 id 19 . . 3 (x = Ax = A)
42, 3eqeq12d 2367 . 2 (x = A → ({x} = x{A} = A))
5 vex 2862 . . 3 x V
65unisn 3907 . 2 {x} = x
74, 6vtoclg 2914 1 (A V{A} = A)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710  {csn 3737  ∪cuni 3891 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-un 3214  df-sn 3741  df-pr 3742  df-uni 3892 This theorem is referenced by:  dfnfc2  3909  dfiota4  4372
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