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| Description: The standard nonnegative integers are a subset of the extended nonnegative integers. (Contributed by AV, 10-Dec-2020.) | 
| Ref | Expression | 
|---|---|
| nn0ssxnn0 | ⊢ ℕ0 ⊆ ℕ0* | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ssun1 4177 | . 2 ⊢ ℕ0 ⊆ (ℕ0 ∪ {+∞}) | |
| 2 | df-xnn0 12602 | . 2 ⊢ ℕ0* = (ℕ0 ∪ {+∞}) | |
| 3 | 1, 2 | sseqtrri 4032 | 1 ⊢ ℕ0 ⊆ ℕ0* | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∪ cun 3948 ⊆ wss 3950 {csn 4625 +∞cpnf 11293 ℕ0cn0 12528 ℕ0*cxnn0 12601 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-v 3481 df-un 3955 df-ss 3967 df-xnn0 12602 | 
| This theorem is referenced by: nn0xnn0 12605 0xnn0 12607 nn0xnn0d 12610 | 
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