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Mirrors > Home > ILE Home > Th. List > xnn0xrnemnf | GIF version |
Description: The extended nonnegative integers are extended reals without negative infinity. (Contributed by AV, 10-Dec-2020.) |
Ref | Expression |
---|---|
xnn0xrnemnf | ⊢ (𝐴 ∈ ℕ0* → (𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xnn0xr 9173 | . 2 ⊢ (𝐴 ∈ ℕ0* → 𝐴 ∈ ℝ*) | |
2 | xnn0nemnf 9179 | . 2 ⊢ (𝐴 ∈ ℕ0* → 𝐴 ≠ -∞) | |
3 | 1, 2 | jca 304 | 1 ⊢ (𝐴 ∈ ℕ0* → (𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 2135 ≠ wne 2334 -∞cmnf 7922 ℝ*cxr 7923 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-in1 604 ax-in2 605 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-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1re 7838 ax-addrcl 7841 ax-rnegex 7853 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-dif 3113 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-mnf 7927 df-xr 7928 df-inn 8849 df-n0 9106 df-xnn0 9169 |
This theorem is referenced by: xnn0xadd0 9794 xnn0add4d 9813 |
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