Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  bj-peano2 Structured version   GIF version

Theorem bj-peano2 7161
Description: Constructive proof of peano2 4245. Temporary note: another possibility is to simply replace sucexg 4174 with bj-sucexg 7145 in the proof of peano2 4245. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-peano2 (A 𝜔 → suc A 𝜔)

Proof of Theorem bj-peano2
StepHypRef Expression
1 bj-omind 7156 . 2 Ind 𝜔
2 bj-indsuc 7151 . 2 (Ind 𝜔 → (A 𝜔 → suc A 𝜔))
31, 2ax-mp 7 1 (A 𝜔 → suc A 𝜔)
Colors of variables: wff set class
Syntax hints:  wi 4   wcel 1374  suc csuc 4051  𝜔com 4240  Ind wind 7149
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 532  ax-in2 533  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-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-nul 3857  ax-pr 3918  ax-un 4120  ax-bd0 7040  ax-bdor 7043  ax-bdex 7046  ax-bdeq 7047  ax-bdel 7048  ax-bdsb 7049  ax-bdsep 7111
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-ral 2289  df-rex 2290  df-rab 2293  df-v 2537  df-dif 2897  df-un 2899  df-nul 3202  df-sn 3356  df-pr 3357  df-uni 3555  df-int 3590  df-suc 4057  df-iom 4241  df-bdc 7068  df-bj-ind 7150
This theorem is referenced by:  peano5set  7162  bj-nn0suc  7182  bj-nn0sucALT  7196
  Copyright terms: Public domain W3C validator