MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  unisn3 Structured version   Visualization version   GIF version

Theorem unisn3 4862
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 4657 . . 3 (𝐴𝐵 → {𝑥𝐵𝑥 = 𝐴} = {𝐴})
21unieqd 4853 . 2 (𝐴𝐵 {𝑥𝐵𝑥 = 𝐴} = {𝐴})
3 unisng 4860 . 2 (𝐴𝐵 {𝐴} = 𝐴)
42, 3eqtrd 2778 1 (𝐴𝐵 {𝑥𝐵𝑥 = 𝐴} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2106  {crab 3068  {csn 4561   cuni 4839
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-un 3892  df-in 3894  df-ss 3904  df-sn 4562  df-pr 4564  df-uni 4840
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator