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Theorem unisn3 4853
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 4655 . . 3 (𝐴𝐵 → {𝑥𝐵𝑥 = 𝐴} = {𝐴})
21unieqd 4846 . 2 (𝐴𝐵 {𝑥𝐵𝑥 = 𝐴} = {𝐴})
3 unisng 4851 . 2 (𝐴𝐵 {𝐴} = 𝐴)
42, 3eqtrd 2860 1 (𝐴𝐵 {𝑥𝐵𝑥 = 𝐴} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1530  wcel 2107  {crab 3146  {csn 4563   cuni 4836
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-rex 3148  df-rab 3151  df-v 3501  df-un 3944  df-sn 4564  df-pr 4566  df-uni 4837
This theorem is referenced by: (None)
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