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Theorem nnpredcl 4622
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 4563) but also holds when it is by uni0 3836. (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 3818 . . . 4 (𝐴 = ∅ → 𝐴 = ∅)
2 uni0 3836 . . . . 5 ∅ = ∅
3 peano1 4593 . . . . 5 ∅ ∈ ω
42, 3eqeltri 2250 . . . 4 ∅ ∈ ω
51, 4eqeltrdi 2268 . . 3 (𝐴 = ∅ → 𝐴 ∈ ω)
65adantl 277 . 2 ((𝐴 ∈ ω ∧ 𝐴 = ∅) → 𝐴 ∈ ω)
7 nnon 4609 . . . . . 6 (𝐴 ∈ ω → 𝐴 ∈ On)
87adantr 276 . . . . 5 ((𝐴 ∈ ω ∧ ∃𝑥 ∈ ω 𝐴 = suc 𝑥) → 𝐴 ∈ On)
9 simpr 110 . . . . 5 ((𝐴 ∈ ω ∧ ∃𝑥 ∈ ω 𝐴 = suc 𝑥) → ∃𝑥 ∈ ω 𝐴 = suc 𝑥)
10 onsucuni2 4563 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐴 = suc 𝑥) → suc 𝐴 = 𝐴)
1110ex 115 . . . . . 6 (𝐴 ∈ On → (𝐴 = suc 𝑥 → suc 𝐴 = 𝐴))
1211rexlimdvw 2598 . . . . 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 109 . . . 4 ((𝐴 ∈ ω ∧ ∃𝑥 ∈ ω 𝐴 = suc 𝑥) → 𝐴 ∈ ω)
1513, 14eqeltrd 2254 . . 3 ((𝐴 ∈ ω ∧ ∃𝑥 ∈ ω 𝐴 = suc 𝑥) → suc 𝐴 ∈ ω)
16 peano2b 4614 . . 3 ( 𝐴 ∈ ω ↔ suc 𝐴 ∈ ω)
1715, 16sylibr 134 . 2 ((𝐴 ∈ ω ∧ ∃𝑥 ∈ ω 𝐴 = suc 𝑥) → 𝐴 ∈ ω)
18 nn0suc 4603 . 2 (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
196, 17, 18mpjaodan 798 1 (𝐴 ∈ ω → 𝐴 ∈ ω)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wcel 2148  wrex 2456  c0 3422   cuni 3809  Oncon0 4363  suc csuc 4365  ωcom 4589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-iinf 4587
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-uni 3810  df-int 3845  df-tr 4102  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590
This theorem is referenced by:  nnpredlt  4623  omp1eomlem  7092  ctmlemr  7106  nnnninfeq2  7126  nninfisollemne  7128  nninfisol  7130  nnsf  14636  peano4nninf  14637
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