ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  rabsn GIF version

Theorem rabsn 3465
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 2116 . . . . 5 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
21pm5.32ri 436 . . . 4 ((𝑥𝐴𝑥 = 𝐵) ↔ (𝐵𝐴𝑥 = 𝐵))
32baib 839 . . 3 (𝐵𝐴 → ((𝑥𝐴𝑥 = 𝐵) ↔ 𝑥 = 𝐵))
43abbidv 2171 . 2 (𝐵𝐴 → {𝑥 ∣ (𝑥𝐴𝑥 = 𝐵)} = {𝑥𝑥 = 𝐵})
5 df-rab 2332 . 2 {𝑥𝐴𝑥 = 𝐵} = {𝑥 ∣ (𝑥𝐴𝑥 = 𝐵)}
6 df-sn 3409 . 2 {𝐵} = {𝑥𝑥 = 𝐵}
74, 5, 63eqtr4g 2113 1 (𝐵𝐴 → {𝑥𝐴𝑥 = 𝐵} = {𝐵})
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1259  wcel 1409  {cab 2042  {crab 2327  {csn 3403
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-rab 2332  df-sn 3409
This theorem is referenced by:  unisn3  4208
  Copyright terms: Public domain W3C validator