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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 4487) but also holds when it is ∅ by uni0 3771. (Contributed by Jim Kingdon, 31-Jul-2022.) |
Ref | Expression |
---|---|
nnpredcl | ⊢ (𝐴 ∈ ω → ∪ 𝐴 ∈ ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unieq 3753 | . . . 4 ⊢ (𝐴 = ∅ → ∪ 𝐴 = ∪ ∅) | |
2 | uni0 3771 | . . . . 5 ⊢ ∪ ∅ = ∅ | |
3 | peano1 4516 | . . . . 5 ⊢ ∅ ∈ ω | |
4 | 2, 3 | eqeltri 2213 | . . . 4 ⊢ ∪ ∅ ∈ ω |
5 | 1, 4 | eqeltrdi 2231 | . . 3 ⊢ (𝐴 = ∅ → ∪ 𝐴 ∈ ω) |
6 | 5 | adantl 275 | . 2 ⊢ ((𝐴 ∈ ω ∧ 𝐴 = ∅) → ∪ 𝐴 ∈ ω) |
7 | nnon 4531 | . . . . . 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 4487 | . . . . . . 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 2556 | . . . . 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 2217 | . . 3 ⊢ ((𝐴 ∈ ω ∧ ∃𝑥 ∈ ω 𝐴 = suc 𝑥) → suc ∪ 𝐴 ∈ ω) |
16 | peano2b 4536 | . . 3 ⊢ (∪ 𝐴 ∈ ω ↔ suc ∪ 𝐴 ∈ ω) | |
17 | 15, 16 | sylibr 133 | . 2 ⊢ ((𝐴 ∈ ω ∧ ∃𝑥 ∈ ω 𝐴 = suc 𝑥) → ∪ 𝐴 ∈ ω) |
18 | nn0suc 4526 | . 2 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)) | |
19 | 6, 17, 18 | mpjaodan 788 | 1 ⊢ (𝐴 ∈ ω → ∪ 𝐴 ∈ ω) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1332 ∈ wcel 1481 ∃wrex 2418 ∅c0 3368 ∪ cuni 3744 Oncon0 4293 suc csuc 4295 ωcom 4512 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-iinf 4510 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-uni 3745 df-int 3780 df-tr 4035 df-iord 4296 df-on 4298 df-suc 4301 df-iom 4513 |
This theorem is referenced by: omp1eomlem 6987 ctmlemr 7001 nnsf 13374 peano4nninf 13375 |
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