Theorem bj-peano2 15879

Description: Constructive proof of peano2 4643. Temporary note: another possibility is to simply replace sucexg 4546 with bj-sucexg 15862 in the proof of peano2 4643. (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

Step	Hyp	Ref	Expression
1		bj-omind 15874	. 2 ⊢ Ind ω
2		bj-indsuc 15868	. 2 ⊢ (Ind ω → (𝐴 ∈ ω → suc 𝐴 ∈ ω))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∈ ω → suc 𝐴 ∈ ω)

Syntax hints:   → wi 4   ∈ wcel 2176  suc csuc 4412  ωcom 4638  Ind wind 15866

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-nul 4170  ax-pr 4253  ax-un 4480  ax-bd0 15753  ax-bdor 15756  ax-bdex 15759  ax-bdeq 15760  ax-bdel 15761  ax-bdsb 15762  ax-bdsep 15824

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-dif 3168  df-un 3170  df-nul 3461  df-sn 3639  df-pr 3640  df-uni 3851  df-int 3886  df-suc 4418  df-iom 4639  df-bdc 15781  df-bj-ind 15867

This theorem is referenced by:  bj-nn0suc  15904  bj-nn0sucALT  15918