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

Theorem sucprc 4115
Description: A proper class is its own successor. (Contributed by NM, 3-Apr-1995.)
Assertion
Ref Expression
sucprc A V → suc A = A)

Proof of Theorem sucprc
StepHypRef Expression
1 df-suc 4074 . . 3 suc A = (A ∪ {A})
2 snprc 3426 . . . 4 A V ↔ {A} = ∅)
3 uneq2 3085 . . . 4 ({A} = ∅ → (A ∪ {A}) = (A ∪ ∅))
42, 3sylbi 114 . . 3 A V → (A ∪ {A}) = (A ∪ ∅))
51, 4syl5eq 2081 . 2 A V → suc A = (A ∪ ∅))
6 un0 3245 . 2 (A ∪ ∅) = A
75, 6syl6eq 2085 1 A V → suc A = A)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1242   wcel 1390  Vcvv 2551  cun 2909  c0 3218  {csn 3367  suc csuc 4068
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-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  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-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-dif 2914  df-un 2916  df-nul 3219  df-sn 3373  df-suc 4074
This theorem is referenced by:  sucprcreg  4227  sucon  4231
  Copyright terms: Public domain W3C validator