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Theorem unisng 3588
 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 3378 . . . 4 (x = A → {x} = {A})
21unieqd 3582 . . 3 (x = A {x} = {A})
3 id 19 . . 3 (x = Ax = A)
42, 3eqeq12d 2051 . 2 (x = A → ( {x} = x {A} = A))
5 vex 2554 . . 3 x V
65unisn 3587 . 2 {x} = x
74, 6vtoclg 2607 1 (A 𝑉 {A} = A)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1242   ∈ wcel 1390  {csn 3367  ∪ 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:  dfnfc2  3589  unisucg  4117  unisn3  4146  opswapg  4750  funfvdm  5179
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