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Theorem nnsuc 7044
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 7040 . . . 4 (𝐴 ∈ ω → ¬ Lim 𝐴)
21adantr 481 . . 3 ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → ¬ Lim 𝐴)
3 nnord 7035 . . . 4 (𝐴 ∈ ω → Ord 𝐴)
4 orduninsuc 7005 . . . . . 6 (Ord 𝐴 → (𝐴 = 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
54adantr 481 . . . . 5 ((Ord 𝐴𝐴 ≠ ∅) → (𝐴 = 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
6 df-lim 5697 . . . . . . 7 (Lim 𝐴 ↔ (Ord 𝐴𝐴 ≠ ∅ ∧ 𝐴 = 𝐴))
76biimpri 218 . . . . . 6 ((Ord 𝐴𝐴 ≠ ∅ ∧ 𝐴 = 𝐴) → Lim 𝐴)
873expia 1264 . . . . 5 ((Ord 𝐴𝐴 ≠ ∅) → (𝐴 = 𝐴 → Lim 𝐴))
95, 8sylbird 250 . . . 4 ((Ord 𝐴𝐴 ≠ ∅) → (¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥 → Lim 𝐴))
103, 9sylan 488 . . 3 ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → (¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥 → Lim 𝐴))
112, 10mt3d 140 . 2 ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ On 𝐴 = suc 𝑥)
12 eleq1 2686 . . . . . . . 8 (𝐴 = suc 𝑥 → (𝐴 ∈ ω ↔ suc 𝑥 ∈ ω))
1312biimpcd 239 . . . . . . 7 (𝐴 ∈ ω → (𝐴 = suc 𝑥 → suc 𝑥 ∈ ω))
14 peano2b 7043 . . . . . . 7 (𝑥 ∈ ω ↔ suc 𝑥 ∈ ω)
1513, 14syl6ibr 242 . . . . . 6 (𝐴 ∈ ω → (𝐴 = suc 𝑥𝑥 ∈ ω))
1615ancrd 576 . . . . 5 (𝐴 ∈ ω → (𝐴 = suc 𝑥 → (𝑥 ∈ ω ∧ 𝐴 = suc 𝑥)))
1716adantld 483 . . . 4 (𝐴 ∈ ω → ((𝑥 ∈ On ∧ 𝐴 = suc 𝑥) → (𝑥 ∈ ω ∧ 𝐴 = suc 𝑥)))
1817reximdv2 3010 . . 3 (𝐴 ∈ ω → (∃𝑥 ∈ On 𝐴 = suc 𝑥 → ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
1918adantr 481 . 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 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wrex 2909  c0 3897   cuni 4409  Ord word 5691  Oncon0 5692  Lim wlim 5693  suc csuc 5694  ωcom 7027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-tr 4723  df-eprel 4995  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-om 7028
This theorem is referenced by:  peano5  7051  nn0suc  7052  inf3lemd  8484  infpssrlem4  9088  fin1a2lem6  9187  bnj158  30558  bnj1098  30615  bnj594  30743
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