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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 12591 | . . 3 ⊢ ℕ0* = (ℕ0 ∪ {+∞}) | |
2 | 1 | eleq2i 2818 | . 2 ⊢ (𝐴 ∈ ℕ0* ↔ 𝐴 ∈ (ℕ0 ∪ {+∞})) |
3 | elun 4145 | . 2 ⊢ (𝐴 ∈ (ℕ0 ∪ {+∞}) ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 ∈ {+∞})) | |
4 | pnfex 11308 | . . . 4 ⊢ +∞ ∈ V | |
5 | 4 | elsn2 4662 | . . 3 ⊢ (𝐴 ∈ {+∞} ↔ 𝐴 = +∞) |
6 | 5 | orbi2i 910 | . 2 ⊢ ((𝐴 ∈ ℕ0 ∨ 𝐴 ∈ {+∞}) ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 = +∞)) |
7 | 2, 3, 6 | 3bitri 296 | 1 ⊢ (𝐴 ∈ ℕ0* ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 = +∞)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∨ wo 845 = wceq 1534 ∈ wcel 2099 ∪ cun 3944 {csn 4623 +∞cpnf 11286 ℕ0cn0 12518 ℕ0*cxnn0 12590 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 ax-sep 5296 ax-pow 5361 ax-un 7738 ax-cnex 11205 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-v 3464 df-un 3951 df-ss 3963 df-pw 4599 df-sn 4624 df-uni 4906 df-pnf 11291 df-xnn0 12591 |
This theorem is referenced by: xnn0xr 12595 pnf0xnn0 12597 xnn0nemnf 12601 xnn0nnn0pnf 12603 xnn0n0n1ge2b 13159 xnn0ge0 13161 xnn0lenn0nn0 13272 xnn0xadd0 13274 xnn0xrge0 13531 tayl0 26386 xnn0gt0 32676 |
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