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Theorem nnpredcl 4580
 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 4521) but also holds when it is ∅ by uni0 3799. (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 3781 . . . 4 (𝐴 = ∅ → 𝐴 = ∅)
2 uni0 3799 . . . . 5 ∅ = ∅
3 peano1 4551 . . . . 5 ∅ ∈ ω
42, 3eqeltri 2230 . . . 4 ∅ ∈ ω
51, 4eqeltrdi 2248 . . 3 (𝐴 = ∅ → 𝐴 ∈ ω)
65adantl 275 . 2 ((𝐴 ∈ ω ∧ 𝐴 = ∅) → 𝐴 ∈ ω)
7 nnon 4567 . . . . . 6 (𝐴 ∈ ω → 𝐴 ∈ On)
87adantr 274 . . . . 5 ((𝐴 ∈ ω ∧ ∃𝑥 ∈ ω 𝐴 = suc 𝑥) → 𝐴 ∈ On)
9 simpr 109 . . . . 5 ((𝐴 ∈ ω ∧ ∃𝑥 ∈ ω 𝐴 = suc 𝑥) → ∃𝑥 ∈ ω 𝐴 = suc 𝑥)
10 onsucuni2 4521 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐴 = suc 𝑥) → suc 𝐴 = 𝐴)
1110ex 114 . . . . . 6 (𝐴 ∈ On → (𝐴 = suc 𝑥 → suc 𝐴 = 𝐴))
1211rexlimdvw 2578 . . . . 5 (𝐴 ∈ On → (∃𝑥 ∈ ω 𝐴 = suc 𝑥 → suc 𝐴 = 𝐴))
138, 9, 12sylc 62 . . . 4 ((𝐴 ∈ ω ∧ ∃𝑥 ∈ ω 𝐴 = suc 𝑥) → suc 𝐴 = 𝐴)
14 simpl 108 . . . 4 ((𝐴 ∈ ω ∧ ∃𝑥 ∈ ω 𝐴 = suc 𝑥) → 𝐴 ∈ ω)
1513, 14eqeltrd 2234 . . 3 ((𝐴 ∈ ω ∧ ∃𝑥 ∈ ω 𝐴 = suc 𝑥) → suc 𝐴 ∈ ω)
16 peano2b 4572 . . 3 ( 𝐴 ∈ ω ↔ suc 𝐴 ∈ ω)
1715, 16sylibr 133 . 2 ((𝐴 ∈ ω ∧ ∃𝑥 ∈ ω 𝐴 = suc 𝑥) → 𝐴 ∈ ω)
18 nn0suc 4561 . 2 (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
196, 17, 18mpjaodan 788 1 (𝐴 ∈ ω → 𝐴 ∈ ω)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1335   ∈ wcel 2128  ∃wrex 2436  ∅c0 3394  ∪ cuni 3772  Oncon0 4322  suc csuc 4324  ωcom 4547 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-nul 4090  ax-pow 4134  ax-pr 4168  ax-un 4392  ax-iinf 4545 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-uni 3773  df-int 3808  df-tr 4063  df-iord 4325  df-on 4327  df-suc 4330  df-iom 4548 This theorem is referenced by:  omp1eomlem  7028  ctmlemr  7042  nnsf  13539  peano4nninf  13540
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