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Theorem rabsn 4400
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 2827 . . . . 5 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
21pm5.32ri 673 . . . 4 ((𝑥𝐴𝑥 = 𝐵) ↔ (𝐵𝐴𝑥 = 𝐵))
32baib 982 . . 3 (𝐵𝐴 → ((𝑥𝐴𝑥 = 𝐵) ↔ 𝑥 = 𝐵))
43abbidv 2879 . 2 (𝐵𝐴 → {𝑥 ∣ (𝑥𝐴𝑥 = 𝐵)} = {𝑥𝑥 = 𝐵})
5 df-rab 3059 . 2 {𝑥𝐴𝑥 = 𝐵} = {𝑥 ∣ (𝑥𝐴𝑥 = 𝐵)}
6 df-sn 4322 . 2 {𝐵} = {𝑥𝑥 = 𝐵}
74, 5, 63eqtr4g 2819 1 (𝐵𝐴 → {𝑥𝐴𝑥 = 𝐵} = {𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  {cab 2746  {crab 3054  {csn 4321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-rab 3059  df-sn 4322
This theorem is referenced by:  unisn3  4605  sylow3lem6  18267  lineunray  32581  pmapat  35570  dia0  36861  nzss  39036  lco0  42744
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