Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > peano2nnd | GIF version |
Description: Peano postulate: a successor of a positive integer is a positive integer. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
nnred.1 | ⊢ (𝜑 → 𝐴 ∈ ℕ) |
Ref | Expression |
---|---|
peano2nnd | ⊢ (𝜑 → (𝐴 + 1) ∈ ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnred.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℕ) | |
2 | peano2nn 8861 | . 2 ⊢ (𝐴 ∈ ℕ → (𝐴 + 1) ∈ ℕ) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → (𝐴 + 1) ∈ ℕ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2135 (class class class)co 5837 1c1 7746 + caddc 7748 ℕcn 8849 |
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-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 ax-sep 4095 ax-cnex 7836 ax-resscn 7837 ax-1re 7839 ax-addrcl 7842 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2724 df-un 3116 df-in 3118 df-ss 3125 df-sn 3577 df-pr 3578 df-op 3580 df-uni 3785 df-int 3820 df-br 3978 df-iota 5148 df-fv 5191 df-ov 5840 df-inn 8850 |
This theorem is referenced by: exp3vallem 10447 bcpasc 10669 caucvgre 10913 resqrexlemdecn 10944 cvgratnnlemmn 11456 cvgratnnlemseq 11457 cvgratnnlemabsle 11458 eftlub 11621 eirraplem 11707 infpnlem1 12278 infpnlem2 12279 1arith 12286 oddennn 12288 exmidunben 12322 nninfdclemp1 12346 nninfdclemlt 12347 cvgcmp2nlemabs 13773 trilpolemeq1 13781 trilpolemlt1 13782 nconstwlpolemgt0 13804 |
Copyright terms: Public domain | W3C validator |