Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 11973 | . 2 ⊢ ℕ0 ⊆ ℕ0* | |
2 | nn0xnn0d.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℕ0) | |
3 | 1, 2 | sseldi 3968 | 1 ⊢ (𝜑 → 𝐴 ∈ ℕ0*) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2113 ℕ0cn0 11900 ℕ0*cxnn0 11970 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-v 3499 df-un 3944 df-in 3946 df-ss 3955 df-xnn0 11971 |
This theorem is referenced by: xnn0xaddcl 12631 fusgrn0eqdrusgr 27355 cusgrrusgr 27366 |
Copyright terms: Public domain | W3C validator |