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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 12600 | . . 3 ⊢ ℕ0* = (ℕ0 ∪ {+∞}) | |
| 2 | 1 | eleq2i 2833 | . 2 ⊢ (𝐴 ∈ ℕ0* ↔ 𝐴 ∈ (ℕ0 ∪ {+∞})) |
| 3 | elun 4153 | . 2 ⊢ (𝐴 ∈ (ℕ0 ∪ {+∞}) ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 ∈ {+∞})) | |
| 4 | pnfex 11314 | . . . 4 ⊢ +∞ ∈ V | |
| 5 | 4 | elsn2 4665 | . . 3 ⊢ (𝐴 ∈ {+∞} ↔ 𝐴 = +∞) |
| 6 | 5 | orbi2i 913 | . 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 206 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ∪ cun 3949 {csn 4626 +∞cpnf 11292 ℕ0cn0 12526 ℕ0*cxnn0 12599 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-pow 5365 ax-un 7755 ax-cnex 11211 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-un 3956 df-ss 3968 df-pw 4602 df-sn 4627 df-uni 4908 df-pnf 11297 df-xnn0 12600 |
| This theorem is referenced by: xnn0xr 12604 pnf0xnn0 12606 xnn0nemnf 12610 xnn0nnn0pnf 12612 xnn0n0n1ge2b 13174 xnn0ge0 13176 xnn0lenn0nn0 13287 xnn0xadd0 13289 xnn0xrge0 13546 tayl0 26403 xnn0gt0 32773 fldextrspundgdvdslem 33730 |
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