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Theorem bj-peano2 11717
Description: Constructive proof of peano2 4408. Temporary note: another possibility is to simply replace sucexg 4313 with bj-sucexg 11696 in the proof of peano2 4408. (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 11712 . 2 Ind ω
2 bj-indsuc 11706 . 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 1438  suc csuc 4190  ωcom 4403  Ind wind 11704
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-nul 3963  ax-pr 4034  ax-un 4258  ax-bd0 11587  ax-bdor 11590  ax-bdex 11593  ax-bdeq 11594  ax-bdel 11595  ax-bdsb 11596  ax-bdsep 11658
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-dif 3001  df-un 3003  df-nul 3287  df-sn 3450  df-pr 3451  df-uni 3652  df-int 3687  df-suc 4196  df-iom 4404  df-bdc 11615  df-bj-ind 11705
This theorem is referenced by:  bj-nn0suc  11742  bj-nn0sucALT  11756
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