Theorem nnpredcl 11890

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 4380) but also holds when it is ∅ by uni0 3680. (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 3662	. . . 4 ⊢ (𝐴 = ∅ → ∪ 𝐴 = ∪ ∅)
2		uni0 3680	. . . . 5 ⊢ ∪ ∅ = ∅
3		peano1 4409	. . . . 5 ⊢ ∅ ∈ ω
4	2, 3	eqeltri 2160	. . . 4 ⊢ ∪ ∅ ∈ ω
5	1, 4	syl6eqel 2178	. . 3 ⊢ (𝐴 = ∅ → ∪ 𝐴 ∈ ω)
6	5	adantl 271	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐴 = ∅) → ∪ 𝐴 ∈ ω)
7		nnon 4424	. . . . . 6 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On)
8	7	adantr 270	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ ∃𝑥 ∈ ω 𝐴 = suc 𝑥) → 𝐴 ∈ On)
9		simpr 108	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ ∃𝑥 ∈ ω 𝐴 = suc 𝑥) → ∃𝑥 ∈ ω 𝐴 = suc 𝑥)
10		onsucuni2 4380	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐴 = suc 𝑥) → suc ∪ 𝐴 = 𝐴)
11	10	ex 113	. . . . . 6 ⊢ (𝐴 ∈ On → (𝐴 = suc 𝑥 → suc ∪ 𝐴 = 𝐴))
12	11	rexlimdvw 2492	. . . . 5 ⊢ (𝐴 ∈ On → (∃𝑥 ∈ ω 𝐴 = suc 𝑥 → suc ∪ 𝐴 = 𝐴))
13	8, 9, 12	sylc 61	. . . 4 ⊢ ((𝐴 ∈ ω ∧ ∃𝑥 ∈ ω 𝐴 = suc 𝑥) → suc ∪ 𝐴 = 𝐴)
14		simpl 107	. . . 4 ⊢ ((𝐴 ∈ ω ∧ ∃𝑥 ∈ ω 𝐴 = suc 𝑥) → 𝐴 ∈ ω)
15	13, 14	eqeltrd 2164	. . 3 ⊢ ((𝐴 ∈ ω ∧ ∃𝑥 ∈ ω 𝐴 = suc 𝑥) → suc ∪ 𝐴 ∈ ω)
16		peano2b 4429	. . 3 ⊢ (∪ 𝐴 ∈ ω ↔ suc ∪ 𝐴 ∈ ω)
17	15, 16	sylibr 132	. 2 ⊢ ((𝐴 ∈ ω ∧ ∃𝑥 ∈ ω 𝐴 = suc 𝑥) → ∪ 𝐴 ∈ ω)
18		nn0suc 4419	. 2 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
19	6, 17, 18	mpjaodan 747	1 ⊢ (𝐴 ∈ ω → ∪ 𝐴 ∈ ω)

Syntax hints:   → wi 4   ∧ wa 102   = wceq 1289   ∈ wcel 1438  ∃wrex 2360  ∅c0 3286  ∪ cuni 3653  Oncon0 4190  suc csuc 4192  ωcom 4405

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-iinf 4403

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-uni 3654  df-int 3689  df-tr 3937  df-iord 4193  df-on 4195  df-suc 4198  df-iom 4406

This theorem is referenced by:  nnsf  11895  peano4nninf  11896