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Theorem rabsn 4721
Description: Condition where a restricted class abstraction is a singleton. (Contributed by NM, 28-May-2006.) (Proof shortened by AV, 26-Aug-2022.)
Assertion
Ref Expression
rabsn (𝐵𝐴 → {𝑥𝐴𝑥 = 𝐵} = {𝐵})
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem rabsn
StepHypRef Expression
1 eleq1 2817 . . . . 5 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
21pm5.32ri 575 . . . 4 ((𝑥𝐴𝑥 = 𝐵) ↔ (𝐵𝐴𝑥 = 𝐵))
32baib 535 . . 3 (𝐵𝐴 → ((𝑥𝐴𝑥 = 𝐵) ↔ 𝑥 = 𝐵))
43alrimiv 1923 . 2 (𝐵𝐴 → ∀𝑥((𝑥𝐴𝑥 = 𝐵) ↔ 𝑥 = 𝐵))
5 rabeqsn 4665 . 2 ({𝑥𝐴𝑥 = 𝐵} = {𝐵} ↔ ∀𝑥((𝑥𝐴𝑥 = 𝐵) ↔ 𝑥 = 𝐵))
64, 5sylibr 233 1 (𝐵𝐴 → {𝑥𝐴𝑥 = 𝐵} = {𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wal 1532   = wceq 1534  wcel 2099  {crab 3428  {csn 4624
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 3429  df-sn 4625
This theorem is referenced by:  unisn3  4926  sylow3lem6  19580  lineunray  35737  pmapat  39230  dia0  40519  nzss  43748  lco0  47489
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