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Mirrors > Home > MPE Home > Th. List > nn0ssxnn0 | Structured version Visualization version GIF version |
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 4150 | . 2 ⊢ ℕ0 ⊆ (ℕ0 ∪ {+∞}) | |
2 | df-xnn0 11971 | . 2 ⊢ ℕ0* = (ℕ0 ∪ {+∞}) | |
3 | 1, 2 | sseqtrri 4006 | 1 ⊢ ℕ0 ⊆ ℕ0* |
Colors of variables: wff setvar class |
Syntax hints: ∪ cun 3936 ⊆ wss 3938 {csn 4569 +∞cpnf 10674 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 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-v 3498 df-un 3943 df-in 3945 df-ss 3954 df-xnn0 11971 |
This theorem is referenced by: nn0xnn0 11974 0xnn0 11976 nn0xnn0d 11979 |
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