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Theorem bj-peano2 13190
 Description: Constructive proof of peano2 4509. Temporary note: another possibility is to simply replace sucexg 4414 with bj-sucexg 13173 in the proof of peano2 4509. (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 13185 . 2 Ind ω
2 bj-indsuc 13179 . 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 1480  suc csuc 4287  ωcom 4504  Ind wind 13177 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-nul 4054  ax-pr 4131  ax-un 4355  ax-bd0 13064  ax-bdor 13067  ax-bdex 13070  ax-bdeq 13071  ax-bdel 13072  ax-bdsb 13073  ax-bdsep 13135 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-dif 3073  df-un 3075  df-nul 3364  df-sn 3533  df-pr 3534  df-uni 3737  df-int 3772  df-suc 4293  df-iom 4505  df-bdc 13092  df-bj-ind 13178 This theorem is referenced by:  bj-nn0suc  13215  bj-nn0sucALT  13229
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