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Theorem bj-peano2 10919
Description: Constructive proof of peano2 4344. Temporary note: another possibility is to simply replace sucexg 4250 with bj-sucexg 10898 in the proof of peano2 4344. (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 10914 . 2 Ind ω
2 bj-indsuc 10908 . 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 1434  suc csuc 4128  ωcom 4339  Ind wind 10906
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-nul 3912  ax-pr 3972  ax-un 4196  ax-bd0 10789  ax-bdor 10792  ax-bdex 10795  ax-bdeq 10796  ax-bdel 10797  ax-bdsb 10798  ax-bdsep 10860
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-rab 2358  df-v 2604  df-dif 2976  df-un 2978  df-nul 3259  df-sn 3412  df-pr 3413  df-uni 3610  df-int 3645  df-suc 4134  df-iom 4340  df-bdc 10817  df-bj-ind 10907
This theorem is referenced by:  bj-nn0suc  10944  bj-nn0sucALT  10958
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