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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 12543 | . 2 ⊢ ℕ0 ⊆ ℕ0* | |
2 | nn0xnn0d.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℕ0) | |
3 | 1, 2 | sselid 3972 | 1 ⊢ (𝜑 → 𝐴 ∈ ℕ0*) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 ℕ0cn0 12468 ℕ0*cxnn0 12540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-v 3468 df-un 3945 df-in 3947 df-ss 3957 df-xnn0 12541 |
This theorem is referenced by: xnn0xaddcl 13210 pcxnn0cl 16791 fusgrn0eqdrusgr 29262 cusgrrusgr 29273 |
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