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

Theorem rabsn 4631
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 2901 . . . . 5 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
21pm5.32ri 579 . . . 4 ((𝑥𝐴𝑥 = 𝐵) ↔ (𝐵𝐴𝑥 = 𝐵))
32baib 539 . . 3 (𝐵𝐴 → ((𝑥𝐴𝑥 = 𝐵) ↔ 𝑥 = 𝐵))
43alrimiv 1928 . 2 (𝐵𝐴 → ∀𝑥((𝑥𝐴𝑥 = 𝐵) ↔ 𝑥 = 𝐵))
5 rabeqsn 4580 . 2 ({𝑥𝐴𝑥 = 𝐵} = {𝐵} ↔ ∀𝑥((𝑥𝐴𝑥 = 𝐵) ↔ 𝑥 = 𝐵))
64, 5sylibr 237 1 (𝐵𝐴 → {𝑥𝐴𝑥 = 𝐵} = {𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1536   = wceq 1538  wcel 2114  {crab 3134  {csn 4539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-rab 3139  df-sn 4540
This theorem is referenced by:  unisn3  4834  sylow3lem6  18748  lineunray  33682  pmapat  37017  dia0  38306  nzss  40955  lco0  44775
  Copyright terms: Public domain W3C validator