![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 2177 | . . 3 ⊢ +∞ = +∞ | |
2 | 1 | olci 732 | . 2 ⊢ (+∞ ∈ ℕ0 ∨ +∞ = +∞) |
3 | elxnn0 9244 | . 2 ⊢ (+∞ ∈ ℕ0* ↔ (+∞ ∈ ℕ0 ∨ +∞ = +∞)) | |
4 | 2, 3 | mpbir 146 | 1 ⊢ +∞ ∈ ℕ0* |
Colors of variables: wff set class |
Syntax hints: ∨ wo 708 = wceq 1353 ∈ wcel 2148 +∞cpnf 7992 ℕ0cn0 9179 ℕ0*cxnn0 9242 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-un 4435 ax-cnex 7905 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-rex 2461 df-v 2741 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-uni 3812 df-pnf 7997 df-xr 7999 df-xnn0 9243 |
This theorem is referenced by: inftonninf 10444 pcxnn0cl 12313 |
Copyright terms: Public domain | W3C validator |