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Theorem nnsuc 7029
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 7025 . . . 4 (𝐴 ∈ ω → ¬ Lim 𝐴)
21adantr 481 . . 3 ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → ¬ Lim 𝐴)
3 nnord 7020 . . . 4 (𝐴 ∈ ω → Ord 𝐴)
4 orduninsuc 6990 . . . . . 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 5687 . . . . . . 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 7028 . . . . . . 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 3008 . . 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 2908  c0 3891   cuni 4402  Ord word 5681  Oncon0 5682  Lim wlim 5683  suc csuc 5684  ωcom 7012
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 4741  ax-nul 4749  ax-pr 4867  ax-un 6902
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 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-tr 4713  df-eprel 4985  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-om 7013
This theorem is referenced by:  peano5  7036  nn0suc  7037  inf3lemd  8468  infpssrlem4  9072  fin1a2lem6  9171  bnj158  30502  bnj1098  30559  bnj594  30687
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