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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-peano2 | GIF version |
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.) |
Ref | Expression |
---|---|
bj-peano2 | ⊢ (𝐴 ∈ ω → suc 𝐴 ∈ ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-omind 15123 | . 2 ⊢ Ind ω | |
2 | bj-indsuc 15117 | . 2 ⊢ (Ind ω → (𝐴 ∈ ω → suc 𝐴 ∈ ω)) | |
3 | 1, 2 | ax-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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