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Mirrors > Home > ILE Home > Th. List > xnn0xr | GIF version |
Description: An extended nonnegative integer is an extended real. (Contributed by AV, 10-Dec-2020.) |
Ref | Expression |
---|---|
xnn0xr | ⊢ (𝐴 ∈ ℕ0* → 𝐴 ∈ ℝ*) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxnn0 9170 | . 2 ⊢ (𝐴 ∈ ℕ0* ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 = +∞)) | |
2 | nn0re 9114 | . . . 4 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℝ) | |
3 | 2 | rexrd 7939 | . . 3 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℝ*) |
4 | pnfxr 7942 | . . . 4 ⊢ +∞ ∈ ℝ* | |
5 | eleq1 2227 | . . . 4 ⊢ (𝐴 = +∞ → (𝐴 ∈ ℝ* ↔ +∞ ∈ ℝ*)) | |
6 | 4, 5 | mpbiri 167 | . . 3 ⊢ (𝐴 = +∞ → 𝐴 ∈ ℝ*) |
7 | 3, 6 | jaoi 706 | . 2 ⊢ ((𝐴 ∈ ℕ0 ∨ 𝐴 = +∞) → 𝐴 ∈ ℝ*) |
8 | 1, 7 | sylbi 120 | 1 ⊢ (𝐴 ∈ ℕ0* → 𝐴 ∈ ℝ*) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 698 = wceq 1342 ∈ wcel 2135 +∞cpnf 7921 ℝ*cxr 7923 ℕ0cn0 9105 ℕ0*cxnn0 9168 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-un 4405 ax-cnex 7835 ax-resscn 7836 ax-1re 7838 ax-addrcl 7841 ax-rnegex 7853 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-uni 3784 df-int 3819 df-pnf 7926 df-xr 7928 df-inn 8849 df-n0 9106 df-xnn0 9169 |
This theorem is referenced by: xnn0xrnemnf 9180 xnn0dcle 9729 xnn0letri 9730 |
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