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Theorem unisng 3826
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 3603 . . . 4 (𝑥 = 𝐴 → {𝑥} = {𝐴})
21unieqd 3820 . . 3 (𝑥 = 𝐴 {𝑥} = {𝐴})
3 id 19 . . 3 (𝑥 = 𝐴𝑥 = 𝐴)
42, 3eqeq12d 2192 . 2 (𝑥 = 𝐴 → ( {𝑥} = 𝑥 {𝐴} = 𝐴))
5 vex 2740 . . 3 𝑥 ∈ V
65unisn 3825 . 2 {𝑥} = 𝑥
74, 6vtoclg 2797 1 (𝐴𝑉 {𝐴} = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  wcel 2148  {csn 3592   cuni 3809
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2739  df-un 3133  df-sn 3598  df-pr 3599  df-uni 3810
This theorem is referenced by:  dfnfc2  3827  unisucg  4413  unisn3  4444  opswapg  5113  funfvdm  5577  en2other2  7191
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