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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 9036 | . 2 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀 + 𝑁) ∈ ℕ0) | |
4 | 1, 2, 3 | mp2an 423 | 1 ⊢ (𝑀 + 𝑁) ∈ ℕ0 |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1481 (class class class)co 5782 + caddc 7647 ℕ0cn0 9001 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-addcom 7744 ax-addass 7746 ax-i2m1 7749 ax-0id 7752 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-br 3938 df-iota 5096 df-fv 5139 df-ov 5785 df-inn 8745 df-n0 9002 |
This theorem is referenced by: numcl 9218 deccl 9220 numsucc 9245 |
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