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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 11816 | . . 3 ⊢ ℕ0* = (ℕ0 ∪ {+∞}) | |
2 | 1 | eleq2i 2874 | . 2 ⊢ (𝐴 ∈ ℕ0* ↔ 𝐴 ∈ (ℕ0 ∪ {+∞})) |
3 | elun 4046 | . 2 ⊢ (𝐴 ∈ (ℕ0 ∪ {+∞}) ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 ∈ {+∞})) | |
4 | pnfex 10540 | . . . 4 ⊢ +∞ ∈ V | |
5 | 4 | elsn2 4509 | . . 3 ⊢ (𝐴 ∈ {+∞} ↔ 𝐴 = +∞) |
6 | 5 | orbi2i 907 | . 2 ⊢ ((𝐴 ∈ ℕ0 ∨ 𝐴 ∈ {+∞}) ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 = +∞)) |
7 | 2, 3, 6 | 3bitri 298 | 1 ⊢ (𝐴 ∈ ℕ0* ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 = +∞)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∨ wo 842 = wceq 1522 ∈ wcel 2081 ∪ cun 3857 {csn 4472 +∞cpnf 10518 ℕ0cn0 11745 ℕ0*cxnn0 11815 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-ext 2769 ax-sep 5094 ax-pow 5157 ax-un 7319 ax-cnex 10439 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-rex 3111 df-v 3439 df-un 3864 df-in 3866 df-ss 3874 df-pw 4455 df-sn 4473 df-uni 4746 df-pnf 10523 df-xnn0 11816 |
This theorem is referenced by: xnn0xr 11820 pnf0xnn0 11822 xnn0nemnf 11826 xnn0nnn0pnf 11828 xnn0n0n1ge2b 12376 xnn0ge0 12378 xnn0lenn0nn0 12488 xnn0xadd0 12490 xnn0xrge0 12741 tayl0 24633 xnn0gt0 30182 |
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