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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 7243 | . . . 4 ⊢ (𝐴 ∈ ω → ¬ Lim 𝐴) | |
2 | 1 | adantr 472 | . . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → ¬ Lim 𝐴) |
3 | nnord 7238 | . . . 4 ⊢ (𝐴 ∈ ω → Ord 𝐴) | |
4 | orduninsuc 7208 | . . . . . 6 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥)) | |
5 | 4 | adantr 472 | . . . . 5 ⊢ ((Ord 𝐴 ∧ 𝐴 ≠ ∅) → (𝐴 = ∪ 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥)) |
6 | df-lim 5889 | . . . . . . 7 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴)) | |
7 | 6 | biimpri 218 | . . . . . 6 ⊢ ((Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴) → Lim 𝐴) |
8 | 7 | 3expia 1115 | . . . . 5 ⊢ ((Ord 𝐴 ∧ 𝐴 ≠ ∅) → (𝐴 = ∪ 𝐴 → Lim 𝐴)) |
9 | 5, 8 | sylbird 250 | . . . 4 ⊢ ((Ord 𝐴 ∧ 𝐴 ≠ ∅) → (¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥 → Lim 𝐴)) |
10 | 3, 9 | sylan 489 | . . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → (¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥 → Lim 𝐴)) |
11 | 2, 10 | mt3d 140 | . 2 ⊢ ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ On 𝐴 = suc 𝑥) |
12 | eleq1 2827 | . . . . . . . 8 ⊢ (𝐴 = suc 𝑥 → (𝐴 ∈ ω ↔ suc 𝑥 ∈ ω)) | |
13 | 12 | biimpcd 239 | . . . . . . 7 ⊢ (𝐴 ∈ ω → (𝐴 = suc 𝑥 → suc 𝑥 ∈ ω)) |
14 | peano2b 7246 | . . . . . . 7 ⊢ (𝑥 ∈ ω ↔ suc 𝑥 ∈ ω) | |
15 | 13, 14 | syl6ibr 242 | . . . . . 6 ⊢ (𝐴 ∈ ω → (𝐴 = suc 𝑥 → 𝑥 ∈ ω)) |
16 | 15 | ancrd 578 | . . . . 5 ⊢ (𝐴 ∈ ω → (𝐴 = suc 𝑥 → (𝑥 ∈ ω ∧ 𝐴 = suc 𝑥))) |
17 | 16 | adantld 484 | . . . 4 ⊢ (𝐴 ∈ ω → ((𝑥 ∈ On ∧ 𝐴 = suc 𝑥) → (𝑥 ∈ ω ∧ 𝐴 = suc 𝑥))) |
18 | 17 | reximdv2 3152 | . . 3 ⊢ (𝐴 ∈ ω → (∃𝑥 ∈ On 𝐴 = suc 𝑥 → ∃𝑥 ∈ ω 𝐴 = suc 𝑥)) |
19 | 18 | adantr 472 | . 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 196 ∧ wa 383 ∧ w3a 1072 = wceq 1632 ∈ wcel 2139 ≠ wne 2932 ∃wrex 3051 ∅c0 4058 ∪ cuni 4588 Ord word 5883 Oncon0 5884 Lim wlim 5885 suc csuc 5886 ωcom 7230 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pr 5055 ax-un 7114 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-rab 3059 df-v 3342 df-sbc 3577 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-br 4805 df-opab 4865 df-tr 4905 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-om 7231 |
This theorem is referenced by: peano5 7254 nn0suc 7255 inf3lemd 8697 infpssrlem4 9320 fin1a2lem6 9419 bnj158 31104 bnj1098 31161 bnj594 31289 |
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