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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 6970 | . . . 4 ⊢ (𝐴 ∈ ω → ¬ Lim 𝐴) | |
2 | 1 | adantr 480 | . . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → ¬ Lim 𝐴) |
3 | nnord 6965 | . . . 4 ⊢ (𝐴 ∈ ω → Ord 𝐴) | |
4 | orduninsuc 6935 | . . . . . 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 5645 | . . . . . . 7 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴)) | |
7 | 6 | biimpri 217 | . . . . . 6 ⊢ ((Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴) → Lim 𝐴) |
8 | 7 | 3expia 1259 | . . . . 5 ⊢ ((Ord 𝐴 ∧ 𝐴 ≠ ∅) → (𝐴 = ∪ 𝐴 → Lim 𝐴)) |
9 | 5, 8 | sylbird 249 | . . . 4 ⊢ ((Ord 𝐴 ∧ 𝐴 ≠ ∅) → (¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥 → Lim 𝐴)) |
10 | 3, 9 | sylan 487 | . . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → (¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥 → Lim 𝐴)) |
11 | 2, 10 | mt3d 139 | . 2 ⊢ ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ On 𝐴 = suc 𝑥) |
12 | eleq1 2676 | . . . . . . . 8 ⊢ (𝐴 = suc 𝑥 → (𝐴 ∈ ω ↔ suc 𝑥 ∈ ω)) | |
13 | 12 | biimpcd 238 | . . . . . . 7 ⊢ (𝐴 ∈ ω → (𝐴 = suc 𝑥 → suc 𝑥 ∈ ω)) |
14 | peano2b 6973 | . . . . . . 7 ⊢ (𝑥 ∈ ω ↔ suc 𝑥 ∈ ω) | |
15 | 13, 14 | syl6ibr 241 | . . . . . 6 ⊢ (𝐴 ∈ ω → (𝐴 = suc 𝑥 → 𝑥 ∈ ω)) |
16 | 15 | ancrd 575 | . . . . 5 ⊢ (𝐴 ∈ ω → (𝐴 = suc 𝑥 → (𝑥 ∈ ω ∧ 𝐴 = suc 𝑥))) |
17 | 16 | adantld 482 | . . . 4 ⊢ (𝐴 ∈ ω → ((𝑥 ∈ On ∧ 𝐴 = suc 𝑥) → (𝑥 ∈ ω ∧ 𝐴 = suc 𝑥))) |
18 | 17 | reximdv2 2997 | . . 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 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∃wrex 2897 ∅c0 3874 ∪ cuni 4372 Ord word 5639 Oncon0 5640 Lim wlim 5641 suc csuc 5642 ωcom 6957 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-tr 4681 df-eprel 4949 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-om 6958 |
This theorem is referenced by: peano5 6981 nn0suc 6982 inf3lemd 8407 infpssrlem4 9011 fin1a2lem6 9110 bnj158 30051 bnj1098 30108 bnj594 30236 |
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