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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 11241 | . . 3 ⊢ ℕ0* = (ℕ0 ∪ {+∞}) | |
2 | 1 | eleq2i 2680 | . 2 ⊢ (𝐴 ∈ ℕ0* ↔ 𝐴 ∈ (ℕ0 ∪ {+∞})) |
3 | elun 3715 | . 2 ⊢ (𝐴 ∈ (ℕ0 ∪ {+∞}) ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 ∈ {+∞})) | |
4 | pnfex 9972 | . . . 4 ⊢ +∞ ∈ V | |
5 | 4 | elsn2 4158 | . . 3 ⊢ (𝐴 ∈ {+∞} ↔ 𝐴 = +∞) |
6 | 5 | orbi2i 540 | . 2 ⊢ ((𝐴 ∈ ℕ0 ∨ 𝐴 ∈ {+∞}) ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 = +∞)) |
7 | 2, 3, 6 | 3bitri 285 | 1 ⊢ (𝐴 ∈ ℕ0* ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 = +∞)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∨ wo 382 = wceq 1475 ∈ wcel 1977 ∪ cun 3538 {csn 4125 +∞cpnf 9950 ℕ0cn0 11169 ℕ0*cxnn0 11240 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-pow 4769 ax-un 6847 ax-cnex 9871 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rex 2902 df-v 3175 df-un 3545 df-in 3547 df-ss 3554 df-pw 4110 df-sn 4126 df-pr 4128 df-uni 4373 df-pnf 9955 df-xr 9957 df-xnn0 11241 |
This theorem is referenced by: xnn0xr 11245 pnf0xnn0 11247 xnn0nemnf 11251 xnn0nnn0pnf 11253 xnn0n0n1ge2b 11841 xnn0ge0 11843 xnn0xadd0 11949 xnn0xrge0 12196 tayl0 23920 |
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