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Theorem rabsn 3599
 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 2203 . . . . 5 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
21pm5.32ri 451 . . . 4 ((𝑥𝐴𝑥 = 𝐵) ↔ (𝐵𝐴𝑥 = 𝐵))
32baib 905 . . 3 (𝐵𝐴 → ((𝑥𝐴𝑥 = 𝐵) ↔ 𝑥 = 𝐵))
43abbidv 2258 . 2 (𝐵𝐴 → {𝑥 ∣ (𝑥𝐴𝑥 = 𝐵)} = {𝑥𝑥 = 𝐵})
5 df-rab 2426 . 2 {𝑥𝐴𝑥 = 𝐵} = {𝑥 ∣ (𝑥𝐴𝑥 = 𝐵)}
6 df-sn 3539 . 2 {𝐵} = {𝑥𝑥 = 𝐵}
74, 5, 63eqtr4g 2198 1 (𝐵𝐴 → {𝑥𝐴𝑥 = 𝐵} = {𝐵})
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1332   ∈ wcel 1481  {cab 2126  {crab 2421  {csn 3533 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-rab 2426  df-sn 3539 This theorem is referenced by:  unisn3  4375
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