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Theorem unisn3 4592
 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 4393 . . 3 (𝐴𝐵 → {𝑥𝐵𝑥 = 𝐴} = {𝐴})
21unieqd 4585 . 2 (𝐴𝐵 {𝑥𝐵𝑥 = 𝐴} = {𝐴})
3 unisng 4591 . 2 (𝐴𝐵 {𝐴} = 𝐴)
42, 3eqtrd 2805 1 (𝐴𝐵 {𝑥𝐵𝑥 = 𝐴} = 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1631   ∈ wcel 2145  {crab 3065  {csn 4317  ∪ cuni 4575 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rex 3067  df-rab 3070  df-v 3353  df-un 3728  df-sn 4318  df-pr 4320  df-uni 4576 This theorem is referenced by: (None)
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