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

Theorem sucidg 4102
Description: Part of Proposition 7.23 of [TakeutiZaring] p. 41 (generalized). (Contributed by NM, 25-Mar-1995.) (Proof shortened by Scott Fenton, 20-Feb-2012.)
Assertion
Ref Expression
sucidg (A 𝑉A suc A)

Proof of Theorem sucidg
StepHypRef Expression
1 eqid 2022 . . 3 A = A
21olci 638 . 2 (A A A = A)
3 elsucg 4090 . 2 (A 𝑉 → (A suc A ↔ (A A A = A)))
42, 3mpbiri 157 1 (A 𝑉A suc A)
Colors of variables: wff set class
Syntax hints:  wi 4   wo 616   = wceq 1228   wcel 1374  suc csuc 4051
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-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-un 2899  df-sn 3356  df-suc 4057
This theorem is referenced by:  sucid  4103  nsuceq0g  4104  trsuc  4109  sucssel  4111  ordsucg  4178  sucunielr  4185  suc11g  4219  nlimsucg  4226  onpsssuc  4231  peano2b  4264  frecsuclem2  5905  bj-peano4  7177
  Copyright terms: Public domain W3C validator