Theorem nnpredcl 4656

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 4597) but also holds when it is ∅ by uni0 3863. (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.

Step	Hyp	Ref	Expression
1		unieq 3845	. . . 4 ⊢ (𝐴 = ∅ → ∪ 𝐴 = ∪ ∅)
2		uni0 3863	. . . . 5 ⊢ ∪ ∅ = ∅
3		peano1 4627	. . . . 5 ⊢ ∅ ∈ ω
4	2, 3	eqeltri 2266	. . . 4 ⊢ ∪ ∅ ∈ ω
5	1, 4	eqeltrdi 2284	. . 3 ⊢ (𝐴 = ∅ → ∪ 𝐴 ∈ ω)
6	5	adantl 277	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐴 = ∅) → ∪ 𝐴 ∈ ω)
7		nnon 4643	. . . . . 6 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On)
8	7	adantr 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 4597	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐴 = suc 𝑥) → suc ∪ 𝐴 = 𝐴)
11	10	ex 115	. . . . . 6 ⊢ (𝐴 ∈ On → (𝐴 = suc 𝑥 → suc ∪ 𝐴 = 𝐴))
12	11	rexlimdvw 2615	. . . . 5 ⊢ (𝐴 ∈ On → (∃𝑥 ∈ ω 𝐴 = suc 𝑥 → suc ∪ 𝐴 = 𝐴))
13	8, 9, 12	sylc 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)
15	13, 14	eqeltrd 2270	. . 3 ⊢ ((𝐴 ∈ ω ∧ ∃𝑥 ∈ ω 𝐴 = suc 𝑥) → suc ∪ 𝐴 ∈ ω)
16		peano2b 4648	. . 3 ⊢ (∪ 𝐴 ∈ ω ↔ suc ∪ 𝐴 ∈ ω)
17	15, 16	sylibr 134	. 2 ⊢ ((𝐴 ∈ ω ∧ ∃𝑥 ∈ ω 𝐴 = suc 𝑥) → ∪ 𝐴 ∈ ω)
18		nn0suc 4637	. 2 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
19	6, 17, 18	mpjaodan 799	1 ⊢ (𝐴 ∈ ω → ∪ 𝐴 ∈ ω)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2164  ∃wrex 2473  ∅c0 3447  ∪ cuni 3836  Oncon0 4395  suc csuc 4397  ωcom 4623

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-iinf 4621

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-uni 3837  df-int 3872  df-tr 4129  df-iord 4398  df-on 4400  df-suc 4403  df-iom 4624

This theorem is referenced by:  nnpredlt  4657  omp1eomlem  7155  ctmlemr  7169  nnnninfeq2  7190  nninfisollemne  7192  nninfisol  7194  nnsf  15565  peano4nninf  15566