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Theorem bj-peano2 15128
Description: Constructive proof of peano2 4609. Temporary note: another possibility is to simply replace sucexg 4512 with bj-sucexg 15111 in the proof of peano2 4609. (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 15123 . 2 Ind ω
2 bj-indsuc 15117 . 2 (Ind ω → (𝐴 ∈ ω → suc 𝐴 ∈ ω))
31, 2ax-mp 5 1 (𝐴 ∈ ω → suc 𝐴 ∈ ω)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 2160  suc csuc 4380  ωcom 4604  Ind wind 15115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-nul 4144  ax-pr 4224  ax-un 4448  ax-bd0 15002  ax-bdor 15005  ax-bdex 15008  ax-bdeq 15009  ax-bdel 15010  ax-bdsb 15011  ax-bdsep 15073
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-dif 3146  df-un 3148  df-nul 3438  df-sn 3613  df-pr 3614  df-uni 3825  df-int 3860  df-suc 4386  df-iom 4605  df-bdc 15030  df-bj-ind 15116
This theorem is referenced by:  bj-nn0suc  15153  bj-nn0sucALT  15167
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