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

Theorem nn0suc 7370
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 2970 . . . 4 (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
2 nnsuc 7362 . . . 4 ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ ω 𝐴 = suc 𝑥)
31, 2sylan2br 588 . . 3 ((𝐴 ∈ ω ∧ ¬ 𝐴 = ∅) → ∃𝑥 ∈ ω 𝐴 = suc 𝑥)
43ex 403 . 2 (𝐴 ∈ ω → (¬ 𝐴 = ∅ → ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
54orrd 852 1 (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 836   = wceq 1601  wcel 2107  wne 2969  wrex 3091  c0 4141  suc csuc 5980  ωcom 7345
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pr 5140  ax-un 7228
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4674  df-br 4889  df-opab 4951  df-tr 4990  df-eprel 5268  df-po 5276  df-so 5277  df-fr 5316  df-we 5318  df-ord 5981  df-on 5982  df-lim 5983  df-suc 5984  df-om 7346
This theorem is referenced by:  nnawordex  8003  nneneq  8433  php  8434  cantnfvalf  8861  cantnflt  8868  hsmexlem9  9584  winainflem  9852  bnj517  31562  trpredlem1  32323  trpred0  32332  trpredrec  32334
  Copyright terms: Public domain W3C validator