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.

Step	Hyp	Ref	Expression
1		unieq 3781	. . . 4 ⊢ (𝐴 = ∅ → ∪ 𝐴 = ∪ ∅)
2		uni0 3799	. . . . 5 ⊢ ∪ ∅ = ∅
3		peano1 4551	. . . . 5 ⊢ ∅ ∈ ω
4	2, 3	eqeltri 2230	. . . 4 ⊢ ∪ ∅ ∈ ω
5	1, 4	eqeltrdi 2248	. . 3 ⊢ (𝐴 = ∅ → ∪ 𝐴 ∈ ω)
6	5	adantl 275	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐴 = ∅) → ∪ 𝐴 ∈ ω)
7		nnon 4567	. . . . . 6 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On)
8	7	adantr 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)
11	10	ex 114	. . . . . 6 ⊢ (𝐴 ∈ On → (𝐴 = suc 𝑥 → suc ∪ 𝐴 = 𝐴))
12	11	rexlimdvw 2578	. . . . 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 108	. . . 4 ⊢ ((𝐴 ∈ ω ∧ ∃𝑥 ∈ ω 𝐴 = suc 𝑥) → 𝐴 ∈ ω)
15	13, 14	eqeltrd 2234	. . 3 ⊢ ((𝐴 ∈ ω ∧ ∃𝑥 ∈ ω 𝐴 = suc 𝑥) → suc ∪ 𝐴 ∈ ω)
16		peano2b 4572	. . 3 ⊢ (∪ 𝐴 ∈ ω ↔ suc ∪ 𝐴 ∈ ω)
17	15, 16	sylibr 133	. 2 ⊢ ((𝐴 ∈ ω ∧ ∃𝑥 ∈ ω 𝐴 = suc 𝑥) → ∪ 𝐴 ∈ ω)
18		nn0suc 4561	. 2 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
19	6, 17, 18	mpjaodan 788	1 ⊢ (𝐴 ∈ ω → ∪ 𝐴 ∈ ω)

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