![]() |
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 8532 | . 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 1445 (class class class)co 5690 1c1 7448 + caddc 7450 ℕcn 8520 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-cnex 7533 ax-resscn 7534 ax-1re 7536 ax-addrcl 7539 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-nf 1402 df-sb 1700 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ral 2375 df-rex 2376 df-v 2635 df-un 3017 df-in 3019 df-ss 3026 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-int 3711 df-br 3868 df-iota 5014 df-fv 5057 df-ov 5693 df-inn 8521 |
This theorem is referenced by: exp3vallem 10087 bcpasc 10305 caucvgre 10545 resqrexlemdecn 10576 cvgratnnlemmn 11083 cvgratnnlemseq 11084 cvgratnnlemabsle 11085 eftlub 11144 eirraplem 11228 oddennn 11647 |
Copyright terms: Public domain | W3C validator |