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

Theorem rabsn 3428
 Description: Condition where a restricted class abstraction is a singleton. (Contributed by NM, 28-May-2006.)
Assertion
Ref Expression
rabsn (B A → {x Ax = B} = {B})
Distinct variable groups:   x,A   x,B

Proof of Theorem rabsn
StepHypRef Expression
1 eleq1 2097 . . . . 5 (x = B → (x AB A))
21pm5.32ri 428 . . . 4 ((x A x = B) ↔ (B A x = B))
32baib 827 . . 3 (B A → ((x A x = B) ↔ x = B))
43abbidv 2152 . 2 (B A → {x ∣ (x A x = B)} = {xx = B})
5 df-rab 2309 . 2 {x Ax = B} = {x ∣ (x A x = B)}
6 df-sn 3373 . 2 {B} = {xx = B}
74, 5, 63eqtr4g 2094 1 (B A → {x Ax = B} = {B})
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242   ∈ wcel 1390  {cab 2023  {crab 2304  {csn 3367 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-rab 2309  df-sn 3373 This theorem is referenced by:  unisn3  4146
 Copyright terms: Public domain W3C validator