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Theorem nnpredcl 11890
Description: The predecessor of a natural number is a natural number. This theorem is most interesting when the natural number is a successor (as seen in theorems like onsucuni2 4380) but also holds when it is by uni0 3680. (Contributed by Jim Kingdon, 31-Jul-2022.)
Assertion
Ref Expression
nnpredcl (𝐴 ∈ ω → 𝐴 ∈ ω)

Proof of Theorem nnpredcl
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 unieq 3662 . . . 4 (𝐴 = ∅ → 𝐴 = ∅)
2 uni0 3680 . . . . 5 ∅ = ∅
3 peano1 4409 . . . . 5 ∅ ∈ ω
42, 3eqeltri 2160 . . . 4 ∅ ∈ ω
51, 4syl6eqel 2178 . . 3 (𝐴 = ∅ → 𝐴 ∈ ω)
65adantl 271 . 2 ((𝐴 ∈ ω ∧ 𝐴 = ∅) → 𝐴 ∈ ω)
7 nnon 4424 . . . . . 6 (𝐴 ∈ ω → 𝐴 ∈ On)
87adantr 270 . . . . 5 ((𝐴 ∈ ω ∧ ∃𝑥 ∈ ω 𝐴 = suc 𝑥) → 𝐴 ∈ On)
9 simpr 108 . . . . 5 ((𝐴 ∈ ω ∧ ∃𝑥 ∈ ω 𝐴 = suc 𝑥) → ∃𝑥 ∈ ω 𝐴 = suc 𝑥)
10 onsucuni2 4380 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐴 = suc 𝑥) → suc 𝐴 = 𝐴)
1110ex 113 . . . . . 6 (𝐴 ∈ On → (𝐴 = suc 𝑥 → suc 𝐴 = 𝐴))
1211rexlimdvw 2492 . . . . 5 (𝐴 ∈ On → (∃𝑥 ∈ ω 𝐴 = suc 𝑥 → suc 𝐴 = 𝐴))
138, 9, 12sylc 61 . . . 4 ((𝐴 ∈ ω ∧ ∃𝑥 ∈ ω 𝐴 = suc 𝑥) → suc 𝐴 = 𝐴)
14 simpl 107 . . . 4 ((𝐴 ∈ ω ∧ ∃𝑥 ∈ ω 𝐴 = suc 𝑥) → 𝐴 ∈ ω)
1513, 14eqeltrd 2164 . . 3 ((𝐴 ∈ ω ∧ ∃𝑥 ∈ ω 𝐴 = suc 𝑥) → suc 𝐴 ∈ ω)
16 peano2b 4429 . . 3 ( 𝐴 ∈ ω ↔ suc 𝐴 ∈ ω)
1715, 16sylibr 132 . 2 ((𝐴 ∈ ω ∧ ∃𝑥 ∈ ω 𝐴 = suc 𝑥) → 𝐴 ∈ ω)
18 nn0suc 4419 . 2 (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
196, 17, 18mpjaodan 747 1 (𝐴 ∈ ω → 𝐴 ∈ ω)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1289  wcel 1438  wrex 2360  c0 3286   cuni 3653  Oncon0 4190  suc csuc 4192  ωcom 4405
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-iinf 4403
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-uni 3654  df-int 3689  df-tr 3937  df-iord 4193  df-on 4195  df-suc 4198  df-iom 4406
This theorem is referenced by:  nnsf  11895  peano4nninf  11896
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