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Mirrors > Home > ILE Home > Th. List > nn0addcli | GIF version |
Description: Closure of addition of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.) |
Ref | Expression |
---|---|
nn0addcl.1 | ⊢ 𝑀 ∈ ℕ0 |
nn0addcl.2 | ⊢ 𝑁 ∈ ℕ0 |
Ref | Expression |
---|---|
nn0addcli | ⊢ (𝑀 + 𝑁) ∈ ℕ0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0addcl.1 | . 2 ⊢ 𝑀 ∈ ℕ0 | |
2 | nn0addcl.2 | . 2 ⊢ 𝑁 ∈ ℕ0 | |
3 | nn0addcl 9005 | . 2 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀 + 𝑁) ∈ ℕ0) | |
4 | 1, 2, 3 | mp2an 422 | 1 ⊢ (𝑀 + 𝑁) ∈ ℕ0 |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1480 (class class class)co 5767 + caddc 7616 ℕ0cn0 8970 |
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 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-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-addcom 7713 ax-addass 7715 ax-i2m1 7718 ax-0id 7721 |
This theorem depends on definitions: df-bi 116 df-3an 964 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-un 3070 df-in 3072 df-ss 3079 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-br 3925 df-iota 5083 df-fv 5126 df-ov 5770 df-inn 8714 df-n0 8971 |
This theorem is referenced by: numcl 9187 deccl 9189 numsucc 9214 |
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