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Mirrors > Home > ILE Home > Th. List > pnf0xnn0 | GIF version |
Description: Positive infinity is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.) |
Ref | Expression |
---|---|
pnf0xnn0 | ⊢ +∞ ∈ ℕ0* |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2115 | . . 3 ⊢ +∞ = +∞ | |
2 | 1 | olci 704 | . 2 ⊢ (+∞ ∈ ℕ0 ∨ +∞ = +∞) |
3 | elxnn0 8946 | . 2 ⊢ (+∞ ∈ ℕ0* ↔ (+∞ ∈ ℕ0 ∨ +∞ = +∞)) | |
4 | 2, 3 | mpbir 145 | 1 ⊢ +∞ ∈ ℕ0* |
Colors of variables: wff set class |
Syntax hints: ∨ wo 680 = wceq 1314 ∈ wcel 1463 +∞cpnf 7721 ℕ0cn0 8881 ℕ0*cxnn0 8944 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-13 1474 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-pow 4058 ax-un 4315 ax-cnex 7636 |
This theorem depends on definitions: df-bi 116 df-tru 1317 df-nf 1420 df-sb 1719 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-rex 2396 df-v 2659 df-un 3041 df-in 3043 df-ss 3050 df-pw 3478 df-sn 3499 df-pr 3500 df-uni 3703 df-pnf 7726 df-xr 7728 df-xnn0 8945 |
This theorem is referenced by: inftonninf 10107 |
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