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Theorem unisng 3676
 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 (𝐴𝑉 {𝐴} = 𝐴)

Proof of Theorem unisng
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 sneq 3461 . . . 4 (𝑥 = 𝐴 → {𝑥} = {𝐴})
21unieqd 3670 . . 3 (𝑥 = 𝐴 {𝑥} = {𝐴})
3 id 19 . . 3 (𝑥 = 𝐴𝑥 = 𝐴)
42, 3eqeq12d 2103 . 2 (𝑥 = 𝐴 → ( {𝑥} = 𝑥 {𝐴} = 𝐴))
5 vex 2623 . . 3 𝑥 ∈ V
65unisn 3675 . 2 {𝑥} = 𝑥
74, 6vtoclg 2680 1 (𝐴𝑉 {𝐴} = 𝐴)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1290   ∈ wcel 1439  {csn 3450  ∪ cuni 3659 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071 This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-rex 2366  df-v 2622  df-un 3004  df-sn 3456  df-pr 3457  df-uni 3660 This theorem is referenced by:  dfnfc2  3677  unisucg  4250  unisn3  4280  opswapg  4930  funfvdm  5380  en2other2  6883
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