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Mirrors > Home > MPE Home > Th. List > nn0xnn0d | Structured version Visualization version 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 12600 | . 2 ⊢ ℕ0 ⊆ ℕ0* | |
2 | nn0xnn0d.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℕ0) | |
3 | 1, 2 | sselid 3993 | 1 ⊢ (𝜑 → 𝐴 ∈ ℕ0*) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ℕ0cn0 12524 ℕ0*cxnn0 12597 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-un 3968 df-ss 3980 df-xnn0 12598 |
This theorem is referenced by: xnn0xaddcl 13274 pcxnn0cl 16894 fusgrn0eqdrusgr 29603 cusgrrusgr 29614 |
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