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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-peano2 | GIF version |
Description: Constructive proof of peano2 4504. Temporary note: another possibility is to simply replace sucexg 4409 with bj-sucexg 13109 in the proof of peano2 4504. (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 13121 | . 2 ⊢ Ind ω | |
2 | bj-indsuc 13115 | . 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 1480 suc csuc 4282 ωcom 4499 Ind wind 13113 |
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 2119 ax-nul 4049 ax-pr 4126 ax-un 4350 ax-bd0 13000 ax-bdor 13003 ax-bdex 13006 ax-bdeq 13007 ax-bdel 13008 ax-bdsb 13009 ax-bdsep 13071 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-dif 3068 df-un 3070 df-nul 3359 df-sn 3528 df-pr 3529 df-uni 3732 df-int 3767 df-suc 4288 df-iom 4500 df-bdc 13028 df-bj-ind 13114 |
This theorem is referenced by: bj-nn0suc 13151 bj-nn0sucALT 13165 |
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