MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elxnn0 Structured version   Visualization version   GIF version

Theorem elxnn0 11403
Description: An extended nonnegative integer is either a standard nonnegative integer or positive infinity. (Contributed by AV, 10-Dec-2020.)
Assertion
Ref Expression
elxnn0 (𝐴 ∈ ℕ0* ↔ (𝐴 ∈ ℕ0𝐴 = +∞))

Proof of Theorem elxnn0
StepHypRef Expression
1 df-xnn0 11402 . . 3 0* = (ℕ0 ∪ {+∞})
21eleq2i 2722 . 2 (𝐴 ∈ ℕ0*𝐴 ∈ (ℕ0 ∪ {+∞}))
3 elun 3786 . 2 (𝐴 ∈ (ℕ0 ∪ {+∞}) ↔ (𝐴 ∈ ℕ0𝐴 ∈ {+∞}))
4 pnfex 10131 . . . 4 +∞ ∈ V
54elsn2 4244 . . 3 (𝐴 ∈ {+∞} ↔ 𝐴 = +∞)
65orbi2i 540 . 2 ((𝐴 ∈ ℕ0𝐴 ∈ {+∞}) ↔ (𝐴 ∈ ℕ0𝐴 = +∞))
72, 3, 63bitri 286 1 (𝐴 ∈ ℕ0* ↔ (𝐴 ∈ ℕ0𝐴 = +∞))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wo 382   = wceq 1523  wcel 2030  cun 3605  {csn 4210  +∞cpnf 10109  0cn0 11330  0*cxnn0 11401
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-pow 4873  ax-un 6991  ax-cnex 10030
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rex 2947  df-v 3233  df-un 3612  df-in 3614  df-ss 3621  df-pw 4193  df-sn 4211  df-pr 4213  df-uni 4469  df-pnf 10114  df-xr 10116  df-xnn0 11402
This theorem is referenced by:  xnn0xr  11406  pnf0xnn0  11408  xnn0nemnf  11412  xnn0nnn0pnf  11414  xnn0n0n1ge2b  12003  xnn0ge0  12005  xnn0lenn0nn0  12113  xnn0xadd0  12115  xnn0xrge0  12363  tayl0  24161
  Copyright terms: Public domain W3C validator