MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nn0suc Structured version   Visualization version   GIF version

Theorem nn0suc 7733
Description: A natural number is either 0 or a successor. (Contributed by NM, 27-May-1998.)
Assertion
Ref Expression
nn0suc (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
Distinct variable group:   𝑥,𝐴

Proof of Theorem nn0suc
StepHypRef Expression
1 df-ne 2944 . . . 4 (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
2 nnsuc 7721 . . . 4 ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ ω 𝐴 = suc 𝑥)
31, 2sylan2br 595 . . 3 ((𝐴 ∈ ω ∧ ¬ 𝐴 = ∅) → ∃𝑥 ∈ ω 𝐴 = suc 𝑥)
43ex 413 . 2 (𝐴 ∈ ω → (¬ 𝐴 = ∅ → ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
54orrd 860 1 (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 844   = wceq 1539  wcel 2106  wne 2943  wrex 3065  c0 4257  suc csuc 6262  ωcom 7703
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-ext 2709  ax-sep 5222  ax-nul 5229  ax-pr 5351  ax-un 7579
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3432  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4258  df-if 4461  df-pw 4536  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4841  df-br 5075  df-opab 5137  df-tr 5192  df-eprel 5491  df-po 5499  df-so 5500  df-fr 5540  df-we 5542  df-ord 6263  df-on 6264  df-lim 6265  df-suc 6266  df-om 7704
This theorem is referenced by:  nnawordex  8456  nneneq  8980  php  8981  nneneqOLD  8992  phpOLD  8993  cantnfvalf  9411  cantnflt  9418  ttrclselem1  9471  ttrclselem2  9472  hsmexlem9  10169  winainflem  10437  bnj517  32851  nnuni  33678
  Copyright terms: Public domain W3C validator