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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 7440 | . . . 4 ⊢ (𝐴 ∈ ω → ¬ Lim 𝐴) | |
2 | 1 | adantr 481 | . . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → ¬ Lim 𝐴) |
3 | nnord 7435 | . . . 4 ⊢ (𝐴 ∈ ω → Ord 𝐴) | |
4 | orduninsuc 7405 | . . . . . 6 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥)) | |
5 | 4 | adantr 481 | . . . . 5 ⊢ ((Ord 𝐴 ∧ 𝐴 ≠ ∅) → (𝐴 = ∪ 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥)) |
6 | df-lim 6063 | . . . . . . 7 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴)) | |
7 | 6 | biimpri 229 | . . . . . 6 ⊢ ((Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴) → Lim 𝐴) |
8 | 7 | 3expia 1112 | . . . . 5 ⊢ ((Ord 𝐴 ∧ 𝐴 ≠ ∅) → (𝐴 = ∪ 𝐴 → Lim 𝐴)) |
9 | 5, 8 | sylbird 261 | . . . 4 ⊢ ((Ord 𝐴 ∧ 𝐴 ≠ ∅) → (¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥 → Lim 𝐴)) |
10 | 3, 9 | sylan 580 | . . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → (¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥 → Lim 𝐴)) |
11 | 2, 10 | mt3d 150 | . 2 ⊢ ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ On 𝐴 = suc 𝑥) |
12 | eleq1 2868 | . . . . . . . 8 ⊢ (𝐴 = suc 𝑥 → (𝐴 ∈ ω ↔ suc 𝑥 ∈ ω)) | |
13 | 12 | biimpcd 250 | . . . . . . 7 ⊢ (𝐴 ∈ ω → (𝐴 = suc 𝑥 → suc 𝑥 ∈ ω)) |
14 | peano2b 7443 | . . . . . . 7 ⊢ (𝑥 ∈ ω ↔ suc 𝑥 ∈ ω) | |
15 | 13, 14 | syl6ibr 253 | . . . . . 6 ⊢ (𝐴 ∈ ω → (𝐴 = suc 𝑥 → 𝑥 ∈ ω)) |
16 | 15 | ancrd 552 | . . . . 5 ⊢ (𝐴 ∈ ω → (𝐴 = suc 𝑥 → (𝑥 ∈ ω ∧ 𝐴 = suc 𝑥))) |
17 | 16 | adantld 491 | . . . 4 ⊢ (𝐴 ∈ ω → ((𝑥 ∈ On ∧ 𝐴 = suc 𝑥) → (𝑥 ∈ ω ∧ 𝐴 = suc 𝑥))) |
18 | 17 | reximdv2 3231 | . . 3 ⊢ (𝐴 ∈ ω → (∃𝑥 ∈ On 𝐴 = suc 𝑥 → ∃𝑥 ∈ ω 𝐴 = suc 𝑥)) |
19 | 18 | adantr 481 | . 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 207 ∧ wa 396 ∧ w3a 1078 = wceq 1520 ∈ wcel 2079 ≠ wne 2982 ∃wrex 3104 ∅c0 4206 ∪ cuni 4739 Ord word 6057 Oncon0 6058 Lim wlim 6059 suc csuc 6060 ωcom 7427 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-sep 5088 ax-nul 5095 ax-pr 5214 ax-un 7310 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1079 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-ral 3108 df-rex 3109 df-rab 3112 df-v 3434 df-sbc 3702 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-pss 3871 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-tp 4471 df-op 4473 df-uni 4740 df-br 4957 df-opab 5019 df-tr 5058 df-eprel 5345 df-po 5354 df-so 5355 df-fr 5394 df-we 5396 df-ord 6061 df-on 6062 df-lim 6063 df-suc 6064 df-om 7428 |
This theorem is referenced by: peano5 7452 nn0suc 7453 inf3lemd 8925 infpssrlem4 9563 fin1a2lem6 9662 bnj158 31572 bnj1098 31628 bnj594 31756 gonar 32203 goalr 32205 satffun 32217 |
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