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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 11366 | . 2 ⊢ ℕ0 ⊆ ℕ0* | |
2 | nn0xnn0d.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℕ0) | |
3 | 1, 2 | sseldi 3601 | 1 ⊢ (𝜑 → 𝐴 ∈ ℕ0*) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1990 ℕ0cn0 11292 ℕ0*cxnn0 11363 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-un 3579 df-in 3581 df-ss 3588 df-xnn0 11364 |
This theorem is referenced by: xnn0xaddcl 12066 fusgrn0eqdrusgr 26466 cusgrrusgr 26477 |
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