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Mirrors > Home > MPE Home > Th. List > elxnn0 | Structured version Visualization version GIF version |
Description: An extended nonnegative integer is either a standard nonnegative integer or positive infinity. (Contributed by AV, 10-Dec-2020.) |
Ref | Expression |
---|---|
elxnn0 | ⊢ (𝐴 ∈ ℕ0* ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 = +∞)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-xnn0 12493 | . . 3 ⊢ ℕ0* = (ℕ0 ∪ {+∞}) | |
2 | 1 | eleq2i 2830 | . 2 ⊢ (𝐴 ∈ ℕ0* ↔ 𝐴 ∈ (ℕ0 ∪ {+∞})) |
3 | elun 4113 | . 2 ⊢ (𝐴 ∈ (ℕ0 ∪ {+∞}) ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 ∈ {+∞})) | |
4 | pnfex 11215 | . . . 4 ⊢ +∞ ∈ V | |
5 | 4 | elsn2 4630 | . . 3 ⊢ (𝐴 ∈ {+∞} ↔ 𝐴 = +∞) |
6 | 5 | orbi2i 912 | . 2 ⊢ ((𝐴 ∈ ℕ0 ∨ 𝐴 ∈ {+∞}) ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 = +∞)) |
7 | 2, 3, 6 | 3bitri 297 | 1 ⊢ (𝐴 ∈ ℕ0* ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 = +∞)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∨ wo 846 = wceq 1542 ∈ wcel 2107 ∪ cun 3913 {csn 4591 +∞cpnf 11193 ℕ0cn0 12420 ℕ0*cxnn0 12492 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 ax-sep 5261 ax-pow 5325 ax-un 7677 ax-cnex 11114 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-v 3450 df-un 3920 df-in 3922 df-ss 3932 df-pw 4567 df-sn 4592 df-uni 4871 df-pnf 11198 df-xnn0 12493 |
This theorem is referenced by: xnn0xr 12497 pnf0xnn0 12499 xnn0nemnf 12503 xnn0nnn0pnf 12505 xnn0n0n1ge2b 13059 xnn0ge0 13061 xnn0lenn0nn0 13171 xnn0xadd0 13173 xnn0xrge0 13430 tayl0 25737 xnn0gt0 31716 |
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