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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-peano2 | GIF version |
Description: Constructive proof of peano2 4408. Temporary note: another possibility is to simply replace sucexg 4313 with bj-sucexg 11696 in the proof of peano2 4408. (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 11712 | . 2 ⊢ Ind ω | |
2 | bj-indsuc 11706 | . 2 ⊢ (Ind ω → (𝐴 ∈ ω → suc 𝐴 ∈ ω)) | |
3 | 1, 2 | ax-mp 7 | 1 ⊢ (𝐴 ∈ ω → suc 𝐴 ∈ ω) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1438 suc csuc 4190 ωcom 4403 Ind wind 11704 |
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 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-nul 3963 ax-pr 4034 ax-un 4258 ax-bd0 11587 ax-bdor 11590 ax-bdex 11593 ax-bdeq 11594 ax-bdel 11595 ax-bdsb 11596 ax-bdsep 11658 |
This theorem depends on definitions: df-bi 115 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 df-rex 2365 df-rab 2368 df-v 2621 df-dif 3001 df-un 3003 df-nul 3287 df-sn 3450 df-pr 3451 df-uni 3652 df-int 3687 df-suc 4196 df-iom 4404 df-bdc 11615 df-bj-ind 11705 |
This theorem is referenced by: bj-nn0suc 11742 bj-nn0sucALT 11756 |
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