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

Theorem rabsn 4411
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 2832 . . . . 5 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
21pm5.32ri 571 . . . 4 ((𝑥𝐴𝑥 = 𝐵) ↔ (𝐵𝐴𝑥 = 𝐵))
32baib 531 . . 3 (𝐵𝐴 → ((𝑥𝐴𝑥 = 𝐵) ↔ 𝑥 = 𝐵))
43alrimiv 2022 . 2 (𝐵𝐴 → ∀𝑥((𝑥𝐴𝑥 = 𝐵) ↔ 𝑥 = 𝐵))
5 rabeqsn 4371 . 2 ({𝑥𝐴𝑥 = 𝐵} = {𝐵} ↔ ∀𝑥((𝑥𝐴𝑥 = 𝐵) ↔ 𝑥 = 𝐵))
64, 5sylibr 225 1 (𝐵𝐴 → {𝑥𝐴𝑥 = 𝐵} = {𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wal 1650   = wceq 1652  wcel 2155  {crab 3059  {csn 4334
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-rab 3064  df-sn 4335
This theorem is referenced by:  unisn3  4613  sylow3lem6  18311  lineunray  32698  pmapat  35719  dia0  37008  nzss  39190  lco0  42885
  Copyright terms: Public domain W3C validator