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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 4563) but also holds when it is ∅ by uni0 3836. (Contributed by Jim Kingdon, 31-Jul-2022.) |
Ref | Expression |
---|---|
nnpredcl | ⊢ (𝐴 ∈ ω → ∪ 𝐴 ∈ ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unieq 3818 | . . . 4 ⊢ (𝐴 = ∅ → ∪ 𝐴 = ∪ ∅) | |
2 | uni0 3836 | . . . . 5 ⊢ ∪ ∅ = ∅ | |
3 | peano1 4593 | . . . . 5 ⊢ ∅ ∈ ω | |
4 | 2, 3 | eqeltri 2250 | . . . 4 ⊢ ∪ ∅ ∈ ω |
5 | 1, 4 | eqeltrdi 2268 | . . 3 ⊢ (𝐴 = ∅ → ∪ 𝐴 ∈ ω) |
6 | 5 | adantl 277 | . 2 ⊢ ((𝐴 ∈ ω ∧ 𝐴 = ∅) → ∪ 𝐴 ∈ ω) |
7 | nnon 4609 | . . . . . 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 4563 | . . . . . . 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 2598 | . . . . 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 2254 | . . 3 ⊢ ((𝐴 ∈ ω ∧ ∃𝑥 ∈ ω 𝐴 = suc 𝑥) → suc ∪ 𝐴 ∈ ω) |
16 | peano2b 4614 | . . 3 ⊢ (∪ 𝐴 ∈ ω ↔ suc ∪ 𝐴 ∈ ω) | |
17 | 15, 16 | sylibr 134 | . 2 ⊢ ((𝐴 ∈ ω ∧ ∃𝑥 ∈ ω 𝐴 = suc 𝑥) → ∪ 𝐴 ∈ ω) |
18 | nn0suc 4603 | . 2 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)) | |
19 | 6, 17, 18 | mpjaodan 798 | 1 ⊢ (𝐴 ∈ ω → ∪ 𝐴 ∈ ω) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 ∃wrex 2456 ∅c0 3422 ∪ cuni 3809 Oncon0 4363 suc csuc 4365 ωcom 4589 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-iinf 4587 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-uni 3810 df-int 3845 df-tr 4102 df-iord 4366 df-on 4368 df-suc 4371 df-iom 4590 |
This theorem is referenced by: nnpredlt 4623 omp1eomlem 7092 ctmlemr 7106 nnnninfeq2 7126 nninfisollemne 7128 nninfisol 7130 nnsf 14636 peano4nninf 14637 |
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