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Theorem unisn3 4931
Description: Union of a singleton in the form of a restricted class abstraction. (Contributed by NM, 3-Jul-2008.)
Assertion
Ref Expression
unisn3 (𝐴𝐵 {𝑥𝐵𝑥 = 𝐴} = 𝐴)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem unisn3
StepHypRef Expression
1 rabsn 4726 . . 3 (𝐴𝐵 → {𝑥𝐵𝑥 = 𝐴} = {𝐴})
21unieqd 4921 . 2 (𝐴𝐵 {𝑥𝐵𝑥 = 𝐴} = {𝐴})
3 unisng 4928 . 2 (𝐴𝐵 {𝐴} = 𝐴)
42, 3eqtrd 2768 1 (𝐴𝐵 {𝑥𝐵𝑥 = 𝐴} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  {crab 3429  {csn 4629   cuni 4908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3430  df-v 3473  df-un 3952  df-in 3954  df-ss 3964  df-sn 4630  df-pr 4632  df-uni 4909
This theorem is referenced by: (None)
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