Theorem xnn0nnn0pnf 9190

Description: An extended nonnegative integer which is not a standard nonnegative integer is positive infinity. (Contributed by AV, 10-Dec-2020.)

Assertion

Ref	Expression
xnn0nnn0pnf	⊢ ((𝑁 ∈ ℕ0* ∧ ¬ 𝑁 ∈ ℕ0) → 𝑁 = +∞)

Proof of Theorem xnn0nnn0pnf

Step	Hyp	Ref	Expression
1		elxnn0 9179	. . 3 ⊢ (𝑁 ∈ ℕ0* ↔ (𝑁 ∈ ℕ0 ∨ 𝑁 = +∞))
2		pm2.53 712	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∨ 𝑁 = +∞) → (¬ 𝑁 ∈ ℕ0 → 𝑁 = +∞))
3	1, 2	sylbi 120	. 2 ⊢ (𝑁 ∈ ℕ0* → (¬ 𝑁 ∈ ℕ0 → 𝑁 = +∞))
4	3	imp 123	1 ⊢ ((𝑁 ∈ ℕ0* ∧ ¬ 𝑁 ∈ ℕ0) → 𝑁 = +∞)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∨ wo 698   = wceq 1343   ∈ wcel 2136  +∞cpnf 7930  ℕ₀cn0 9114  ℕ₀^*cxnn0 9177

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-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-un 4411  ax-cnex 7844

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-uni 3790  df-pnf 7935  df-xr 7937  df-xnn0 9178

This theorem is referenced by: (None)