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Theorem nnsuc 7247
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 7243 . . . 4 (𝐴 ∈ ω → ¬ Lim 𝐴)
21adantr 472 . . 3 ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → ¬ Lim 𝐴)
3 nnord 7238 . . . 4 (𝐴 ∈ ω → Ord 𝐴)
4 orduninsuc 7208 . . . . . 6 (Ord 𝐴 → (𝐴 = 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
54adantr 472 . . . . 5 ((Ord 𝐴𝐴 ≠ ∅) → (𝐴 = 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
6 df-lim 5889 . . . . . . 7 (Lim 𝐴 ↔ (Ord 𝐴𝐴 ≠ ∅ ∧ 𝐴 = 𝐴))
76biimpri 218 . . . . . 6 ((Ord 𝐴𝐴 ≠ ∅ ∧ 𝐴 = 𝐴) → Lim 𝐴)
873expia 1115 . . . . 5 ((Ord 𝐴𝐴 ≠ ∅) → (𝐴 = 𝐴 → Lim 𝐴))
95, 8sylbird 250 . . . 4 ((Ord 𝐴𝐴 ≠ ∅) → (¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥 → Lim 𝐴))
103, 9sylan 489 . . 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 2827 . . . . . . . 8 (𝐴 = suc 𝑥 → (𝐴 ∈ ω ↔ suc 𝑥 ∈ ω))
1312biimpcd 239 . . . . . . 7 (𝐴 ∈ ω → (𝐴 = suc 𝑥 → suc 𝑥 ∈ ω))
14 peano2b 7246 . . . . . . 7 (𝑥 ∈ ω ↔ suc 𝑥 ∈ ω)
1513, 14syl6ibr 242 . . . . . 6 (𝐴 ∈ ω → (𝐴 = suc 𝑥𝑥 ∈ ω))
1615ancrd 578 . . . . 5 (𝐴 ∈ ω → (𝐴 = suc 𝑥 → (𝑥 ∈ ω ∧ 𝐴 = suc 𝑥)))
1716adantld 484 . . . 4 (𝐴 ∈ ω → ((𝑥 ∈ On ∧ 𝐴 = suc 𝑥) → (𝑥 ∈ ω ∧ 𝐴 = suc 𝑥)))
1817reximdv2 3152 . . 3 (𝐴 ∈ ω → (∃𝑥 ∈ On 𝐴 = suc 𝑥 → ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
1918adantr 472 . 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 383  w3a 1072   = wceq 1632  wcel 2139  wne 2932  wrex 3051  c0 4058   cuni 4588  Ord word 5883  Oncon0 5884  Lim wlim 5885  suc csuc 5886  ωcom 7230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-tr 4905  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-om 7231
This theorem is referenced by:  peano5  7254  nn0suc  7255  inf3lemd  8697  infpssrlem4  9320  fin1a2lem6  9419  bnj158  31104  bnj1098  31161  bnj594  31289
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