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Mirrors > Home > MPE Home > Th. List > 0xnn0 | Structured version Visualization version GIF version |
Description: Zero is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.) |
Ref | Expression |
---|---|
0xnn0 | ⊢ 0 ∈ ℕ0* |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0ssxnn0 12130 | . 2 ⊢ ℕ0 ⊆ ℕ0* | |
2 | 0nn0 12070 | . 2 ⊢ 0 ∈ ℕ0 | |
3 | 1, 2 | sselii 3884 | 1 ⊢ 0 ∈ ℕ0* |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2112 0cc0 10694 ℕ0cn0 12055 ℕ0*cxnn0 12127 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-mulcl 10756 ax-i2m1 10762 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-v 3400 df-un 3858 df-in 3860 df-ss 3870 df-sn 4528 df-n0 12056 df-xnn0 12128 |
This theorem is referenced by: 0edg0rgr 27614 rgrusgrprc 27631 rusgrprc 27632 rgrprcx 27634 |
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