Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  class2seteq Structured version   GIF version

Theorem class2seteq 3907
 Description: Equality theorem for classes and sets . (Contributed by NM, 13-Dec-2005.) (Proof shortened by Raph Levien, 30-Jun-2006.)
Assertion
Ref Expression
class2seteq (A 𝑉 → {x AA V} = A)
Distinct variable group:   x,A
Allowed substitution hint:   𝑉(x)

Proof of Theorem class2seteq
StepHypRef Expression
1 elex 2560 . 2 (A 𝑉A V)
2 ax-1 5 . . . . 5 (A V → (x AA V))
32ralrimiv 2385 . . . 4 (A V → x A A V)
4 rabid2 2480 . . . 4 (A = {x AA V} ↔ x A A V)
53, 4sylibr 137 . . 3 (A V → A = {x AA V})
65eqcomd 2042 . 2 (A V → {x AA V} = A)
71, 6syl 14 1 (A 𝑉 → {x AA V} = A)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1242   ∈ wcel 1390  ∀wral 2300  {crab 2304  Vcvv 2551 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-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-ral 2305  df-rab 2309  df-v 2553 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator