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Mirrors > Home > ILE Home > Th. List > nnpredcl | GIF version |
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 4474) but also holds when it is ∅ by uni0 3758. (Contributed by Jim Kingdon, 31-Jul-2022.) |
Ref | Expression |
---|---|
nnpredcl | ⊢ (𝐴 ∈ ω → ∪ 𝐴 ∈ ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unieq 3740 | . . . 4 ⊢ (𝐴 = ∅ → ∪ 𝐴 = ∪ ∅) | |
2 | uni0 3758 | . . . . 5 ⊢ ∪ ∅ = ∅ | |
3 | peano1 4503 | . . . . 5 ⊢ ∅ ∈ ω | |
4 | 2, 3 | eqeltri 2210 | . . . 4 ⊢ ∪ ∅ ∈ ω |
5 | 1, 4 | eqeltrdi 2228 | . . 3 ⊢ (𝐴 = ∅ → ∪ 𝐴 ∈ ω) |
6 | 5 | adantl 275 | . 2 ⊢ ((𝐴 ∈ ω ∧ 𝐴 = ∅) → ∪ 𝐴 ∈ ω) |
7 | nnon 4518 | . . . . . 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 4474 | . . . . . . 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 2551 | . . . . 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 2214 | . . 3 ⊢ ((𝐴 ∈ ω ∧ ∃𝑥 ∈ ω 𝐴 = suc 𝑥) → suc ∪ 𝐴 ∈ ω) |
16 | peano2b 4523 | . . 3 ⊢ (∪ 𝐴 ∈ ω ↔ suc ∪ 𝐴 ∈ ω) | |
17 | 15, 16 | sylibr 133 | . 2 ⊢ ((𝐴 ∈ ω ∧ ∃𝑥 ∈ ω 𝐴 = suc 𝑥) → ∪ 𝐴 ∈ ω) |
18 | nn0suc 4513 | . 2 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)) | |
19 | 6, 17, 18 | mpjaodan 787 | 1 ⊢ (𝐴 ∈ ω → ∪ 𝐴 ∈ ω) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 ∃wrex 2415 ∅c0 3358 ∪ cuni 3731 Oncon0 4280 suc csuc 4282 ωcom 4499 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-iinf 4497 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-uni 3732 df-int 3767 df-tr 4022 df-iord 4283 df-on 4285 df-suc 4288 df-iom 4500 |
This theorem is referenced by: omp1eomlem 6972 ctmlemr 6986 nnsf 13188 peano4nninf 13189 |
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