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Theorem nn0suc 7836
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 2933 . . . 4 (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
2 nnsuc 7826 . . . 4 ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ ω 𝐴 = suc 𝑥)
31, 2sylan2br 595 . . 3 ((𝐴 ∈ ω ∧ ¬ 𝐴 = ∅) → ∃𝑥 ∈ ω 𝐴 = suc 𝑥)
43ex 412 . 2 (𝐴 ∈ ω → (¬ 𝐴 = ∅ → ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
54orrd 863 1 (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 847   = wceq 1541  wcel 2113  wne 2932  wrex 3060  c0 4285  suc csuc 6319  ωcom 7808
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-tr 5206  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-om 7809
This theorem is referenced by:  nnawordex  8565  nnaordex2  8567  nneneq  9130  php  9131  cantnfvalf  9574  cantnflt  9581  ttrclselem1  9634  ttrclselem2  9635  hsmexlem9  10335  winainflem  10604  bnj517  35041  nnuni  35921
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