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Mirrors > Home > MPE Home > Th. List > nnsuc | Structured version Visualization version GIF version |
Description: A nonzero natural number is a successor. (Contributed by NM, 18-Feb-2004.) |
Ref | Expression |
---|---|
nnsuc | ⊢ ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ ω 𝐴 = suc 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnlim 7862 | . . . 4 ⊢ (𝐴 ∈ ω → ¬ Lim 𝐴) | |
2 | 1 | adantr 480 | . . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → ¬ Lim 𝐴) |
3 | nnord 7856 | . . . 4 ⊢ (𝐴 ∈ ω → Ord 𝐴) | |
4 | orduninsuc 7825 | . . . . . 6 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥)) | |
5 | 4 | adantr 480 | . . . . 5 ⊢ ((Ord 𝐴 ∧ 𝐴 ≠ ∅) → (𝐴 = ∪ 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥)) |
6 | df-lim 6359 | . . . . . . 7 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴)) | |
7 | 6 | biimpri 227 | . . . . . 6 ⊢ ((Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴) → Lim 𝐴) |
8 | 7 | 3expia 1118 | . . . . 5 ⊢ ((Ord 𝐴 ∧ 𝐴 ≠ ∅) → (𝐴 = ∪ 𝐴 → Lim 𝐴)) |
9 | 5, 8 | sylbird 260 | . . . 4 ⊢ ((Ord 𝐴 ∧ 𝐴 ≠ ∅) → (¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥 → Lim 𝐴)) |
10 | 3, 9 | sylan 579 | . . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → (¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥 → Lim 𝐴)) |
11 | 2, 10 | mt3d 148 | . 2 ⊢ ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ On 𝐴 = suc 𝑥) |
12 | eleq1 2813 | . . . . . . . 8 ⊢ (𝐴 = suc 𝑥 → (𝐴 ∈ ω ↔ suc 𝑥 ∈ ω)) | |
13 | 12 | biimpcd 248 | . . . . . . 7 ⊢ (𝐴 ∈ ω → (𝐴 = suc 𝑥 → suc 𝑥 ∈ ω)) |
14 | peano2b 7865 | . . . . . . 7 ⊢ (𝑥 ∈ ω ↔ suc 𝑥 ∈ ω) | |
15 | 13, 14 | imbitrrdi 251 | . . . . . 6 ⊢ (𝐴 ∈ ω → (𝐴 = suc 𝑥 → 𝑥 ∈ ω)) |
16 | 15 | ancrd 551 | . . . . 5 ⊢ (𝐴 ∈ ω → (𝐴 = suc 𝑥 → (𝑥 ∈ ω ∧ 𝐴 = suc 𝑥))) |
17 | 16 | adantld 490 | . . . 4 ⊢ (𝐴 ∈ ω → ((𝑥 ∈ On ∧ 𝐴 = suc 𝑥) → (𝑥 ∈ ω ∧ 𝐴 = suc 𝑥))) |
18 | 17 | reximdv2 3156 | . . 3 ⊢ (𝐴 ∈ ω → (∃𝑥 ∈ On 𝐴 = suc 𝑥 → ∃𝑥 ∈ ω 𝐴 = suc 𝑥)) |
19 | 18 | adantr 480 | . 2 ⊢ ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → (∃𝑥 ∈ On 𝐴 = suc 𝑥 → ∃𝑥 ∈ ω 𝐴 = suc 𝑥)) |
20 | 11, 19 | mpd 15 | 1 ⊢ ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ ω 𝐴 = suc 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2932 ∃wrex 3062 ∅c0 4314 ∪ cuni 4899 Ord word 6353 Oncon0 6354 Lim wlim 6355 suc csuc 6356 ωcom 7848 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pr 5417 ax-un 7718 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-tr 5256 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-om 7849 |
This theorem is referenced by: peano5 7877 peano5OLD 7878 nn0suc 7879 inf3lemd 9618 infpssrlem4 10297 fin1a2lem6 10396 bnj158 34229 bnj1098 34283 bnj594 34412 gonar 34875 goalr 34877 satffun 34889 |
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