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Theorem nn0suc 7905
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 2937 . . . 4 (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
2 nnsuc 7892 . . . 4 ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ ω 𝐴 = suc 𝑥)
31, 2sylan2br 593 . . 3 ((𝐴 ∈ ω ∧ ¬ 𝐴 = ∅) → ∃𝑥 ∈ ω 𝐴 = suc 𝑥)
43ex 411 . 2 (𝐴 ∈ ω → (¬ 𝐴 = ∅ → ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
54orrd 861 1 (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 845   = wceq 1533  wcel 2098  wne 2936  wrex 3066  c0 4324  suc csuc 6374  ωcom 7874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431  ax-un 7744
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2705  df-cleq 2719  df-clel 2805  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-opab 5213  df-tr 5268  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5635  df-we 5637  df-ord 6375  df-on 6376  df-lim 6377  df-suc 6378  df-om 7875
This theorem is referenced by:  nnawordex  8662  nnaordex2  8664  nneneq  9238  php  9239  nneneqOLD  9250  phpOLD  9251  cantnfvalf  9694  cantnflt  9701  ttrclselem1  9754  ttrclselem2  9755  hsmexlem9  10454  winainflem  10722  bnj517  34521  nnuni  35326
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