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Theorem rabsn 4200
 Description: Condition where a restricted class abstraction is a singleton. (Contributed by NM, 28-May-2006.)
Assertion
Ref Expression
rabsn (𝐵𝐴 → {𝑥𝐴𝑥 = 𝐵} = {𝐵})
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem rabsn
StepHypRef Expression
1 eleq1 2676 . . . . 5 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
21pm5.32ri 668 . . . 4 ((𝑥𝐴𝑥 = 𝐵) ↔ (𝐵𝐴𝑥 = 𝐵))
32baib 942 . . 3 (𝐵𝐴 → ((𝑥𝐴𝑥 = 𝐵) ↔ 𝑥 = 𝐵))
43abbidv 2728 . 2 (𝐵𝐴 → {𝑥 ∣ (𝑥𝐴𝑥 = 𝐵)} = {𝑥𝑥 = 𝐵})
5 df-rab 2905 . 2 {𝑥𝐴𝑥 = 𝐵} = {𝑥 ∣ (𝑥𝐴𝑥 = 𝐵)}
6 df-sn 4126 . 2 {𝐵} = {𝑥𝑥 = 𝐵}
74, 5, 63eqtr4g 2669 1 (𝐵𝐴 → {𝑥𝐴𝑥 = 𝐵} = {𝐵})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  {crab 2900  {csn 4125 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-rab 2905  df-sn 4126 This theorem is referenced by:  unisn3  4389  sylow3lem6  17870  lineunray  31424  pmapat  34067  dia0  35359  nzss  37538  lco0  42010
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