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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 11402 | . . 3 ⊢ ℕ0* = (ℕ0 ∪ {+∞}) | |
2 | 1 | eleq2i 2722 | . 2 ⊢ (𝐴 ∈ ℕ0* ↔ 𝐴 ∈ (ℕ0 ∪ {+∞})) |
3 | elun 3786 | . 2 ⊢ (𝐴 ∈ (ℕ0 ∪ {+∞}) ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 ∈ {+∞})) | |
4 | pnfex 10131 | . . . 4 ⊢ +∞ ∈ V | |
5 | 4 | elsn2 4244 | . . 3 ⊢ (𝐴 ∈ {+∞} ↔ 𝐴 = +∞) |
6 | 5 | orbi2i 540 | . 2 ⊢ ((𝐴 ∈ ℕ0 ∨ 𝐴 ∈ {+∞}) ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 = +∞)) |
7 | 2, 3, 6 | 3bitri 286 | 1 ⊢ (𝐴 ∈ ℕ0* ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 = +∞)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∨ wo 382 = wceq 1523 ∈ wcel 2030 ∪ cun 3605 {csn 4210 +∞cpnf 10109 ℕ0cn0 11330 ℕ0*cxnn0 11401 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-pow 4873 ax-un 6991 ax-cnex 10030 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-rex 2947 df-v 3233 df-un 3612 df-in 3614 df-ss 3621 df-pw 4193 df-sn 4211 df-pr 4213 df-uni 4469 df-pnf 10114 df-xr 10116 df-xnn0 11402 |
This theorem is referenced by: xnn0xr 11406 pnf0xnn0 11408 xnn0nemnf 11412 xnn0nnn0pnf 11414 xnn0n0n1ge2b 12003 xnn0ge0 12005 xnn0lenn0nn0 12113 xnn0xadd0 12115 xnn0xrge0 12363 tayl0 24161 |
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