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Mirrors > Home > ILE Home > Th. List > nn0xnn0d | GIF version |
Description: A standard nonnegative integer is an extended nonnegative integer, deduction form. (Contributed by AV, 10-Dec-2020.) |
Ref | Expression |
---|---|
nn0xnn0d.1 | ⊢ (𝜑 → 𝐴 ∈ ℕ0) |
Ref | Expression |
---|---|
nn0xnn0d | ⊢ (𝜑 → 𝐴 ∈ ℕ0*) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0ssxnn0 9067 | . 2 ⊢ ℕ0 ⊆ ℕ0* | |
2 | nn0xnn0d.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℕ0) | |
3 | 1, 2 | sseldi 3100 | 1 ⊢ (𝜑 → 𝐴 ∈ ℕ0*) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1481 ℕ0cn0 9001 ℕ0*cxnn0 9064 |
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 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-xnn0 9065 |
This theorem is referenced by: (None) |
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