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Theorem nnpredcl 4504
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 4447) but also holds when it is by uni0 3731. (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 3713 . . . 4 (𝐴 = ∅ → 𝐴 = ∅)
2 uni0 3731 . . . . 5 ∅ = ∅
3 peano1 4476 . . . . 5 ∅ ∈ ω
42, 3eqeltri 2188 . . . 4 ∅ ∈ ω
51, 4syl6eqel 2206 . . 3 (𝐴 = ∅ → 𝐴 ∈ ω)
65adantl 273 . 2 ((𝐴 ∈ ω ∧ 𝐴 = ∅) → 𝐴 ∈ ω)
7 nnon 4491 . . . . . 6 (𝐴 ∈ ω → 𝐴 ∈ On)
87adantr 272 . . . . 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 4447 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐴 = suc 𝑥) → suc 𝐴 = 𝐴)
1110ex 114 . . . . . 6 (𝐴 ∈ On → (𝐴 = suc 𝑥 → suc 𝐴 = 𝐴))
1211rexlimdvw 2528 . . . . 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 2192 . . 3 ((𝐴 ∈ ω ∧ ∃𝑥 ∈ ω 𝐴 = suc 𝑥) → suc 𝐴 ∈ ω)
16 peano2b 4496 . . 3 ( 𝐴 ∈ ω ↔ suc 𝐴 ∈ ω)
1715, 16sylibr 133 . 2 ((𝐴 ∈ ω ∧ ∃𝑥 ∈ ω 𝐴 = suc 𝑥) → 𝐴 ∈ ω)
18 nn0suc 4486 . 2 (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
196, 17, 18mpjaodan 770 1 (𝐴 ∈ ω → 𝐴 ∈ ω)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1314  wcel 1463  wrex 2392  c0 3331   cuni 3704  Oncon0 4253  suc csuc 4255  ωcom 4472
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-iinf 4470
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-uni 3705  df-int 3740  df-tr 3995  df-iord 4256  df-on 4258  df-suc 4261  df-iom 4473
This theorem is referenced by:  omp1eomlem  6945  ctmlemr  6959  nnsf  13033  peano4nninf  13034
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