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Theorem nnsuc 7866
Description: A nonzero natural number is a successor. (Contributed by NM, 18-Feb-2004.)
Assertion
Ref Expression
nnsuc ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ ω 𝐴 = suc 𝑥)
Distinct variable group:   𝑥,𝐴

Proof of Theorem nnsuc
StepHypRef Expression
1 nnlim 7862 . . . 4 (𝐴 ∈ ω → ¬ Lim 𝐴)
21adantr 480 . . 3 ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → ¬ Lim 𝐴)
3 nnord 7856 . . . 4 (𝐴 ∈ ω → Ord 𝐴)
4 orduninsuc 7825 . . . . . 6 (Ord 𝐴 → (𝐴 = 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
54adantr 480 . . . . 5 ((Ord 𝐴𝐴 ≠ ∅) → (𝐴 = 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
6 df-lim 6359 . . . . . . 7 (Lim 𝐴 ↔ (Ord 𝐴𝐴 ≠ ∅ ∧ 𝐴 = 𝐴))
76biimpri 227 . . . . . 6 ((Ord 𝐴𝐴 ≠ ∅ ∧ 𝐴 = 𝐴) → Lim 𝐴)
873expia 1118 . . . . 5 ((Ord 𝐴𝐴 ≠ ∅) → (𝐴 = 𝐴 → Lim 𝐴))
95, 8sylbird 260 . . . 4 ((Ord 𝐴𝐴 ≠ ∅) → (¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥 → Lim 𝐴))
103, 9sylan 579 . . 3 ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → (¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥 → Lim 𝐴))
112, 10mt3d 148 . 2 ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ On 𝐴 = suc 𝑥)
12 eleq1 2813 . . . . . . . 8 (𝐴 = suc 𝑥 → (𝐴 ∈ ω ↔ suc 𝑥 ∈ ω))
1312biimpcd 248 . . . . . . 7 (𝐴 ∈ ω → (𝐴 = suc 𝑥 → suc 𝑥 ∈ ω))
14 peano2b 7865 . . . . . . 7 (𝑥 ∈ ω ↔ suc 𝑥 ∈ ω)
1513, 14imbitrrdi 251 . . . . . 6 (𝐴 ∈ ω → (𝐴 = suc 𝑥𝑥 ∈ ω))
1615ancrd 551 . . . . 5 (𝐴 ∈ ω → (𝐴 = suc 𝑥 → (𝑥 ∈ ω ∧ 𝐴 = suc 𝑥)))
1716adantld 490 . . . 4 (𝐴 ∈ ω → ((𝑥 ∈ On ∧ 𝐴 = suc 𝑥) → (𝑥 ∈ ω ∧ 𝐴 = suc 𝑥)))
1817reximdv2 3156 . . 3 (𝐴 ∈ ω → (∃𝑥 ∈ On 𝐴 = suc 𝑥 → ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
1918adantr 480 . 2 ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → (∃𝑥 ∈ On 𝐴 = suc 𝑥 → ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
2011, 19mpd 15 1 ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ ω 𝐴 = suc 𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395  w3a 1084   = wceq 1533  wcel 2098  wne 2932  wrex 3062  c0 4314   cuni 4899  Ord word 6353  Oncon0 6354  Lim wlim 6355  suc csuc 6356  ωcom 7848
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 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-tr 5256  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-om 7849
This theorem is referenced by:  peano5  7877  peano5OLD  7878  nn0suc  7879  inf3lemd  9618  infpssrlem4  10297  fin1a2lem6  10396  bnj158  34229  bnj1098  34283  bnj594  34412  gonar  34875  goalr  34877  satffun  34889
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