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Theorem unisng 3753
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 3538 . . . 4 (𝑥 = 𝐴 → {𝑥} = {𝐴})
21unieqd 3747 . . 3 (𝑥 = 𝐴 {𝑥} = {𝐴})
3 id 19 . . 3 (𝑥 = 𝐴𝑥 = 𝐴)
42, 3eqeq12d 2154 . 2 (𝑥 = 𝐴 → ( {𝑥} = 𝑥 {𝐴} = 𝐴))
5 vex 2689 . . 3 𝑥 ∈ V
65unisn 3752 . 2 {𝑥} = 𝑥
74, 6vtoclg 2746 1 (𝐴𝑉 {𝐴} = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1331  wcel 1480  {csn 3527   cuni 3736
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-uni 3737
This theorem is referenced by:  dfnfc2  3754  unisucg  4336  unisn3  4366  opswapg  5025  funfvdm  5484  en2other2  7057
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