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Mirrors > Home > MPE Home > Th. List > dfnn3 | Structured version Visualization version GIF version |
Description: Alternate definition of the set of positive integers. Definition of positive integers in [Apostol] p. 22. (Contributed by NM, 3-Jul-2005.) |
Ref | Expression |
---|---|
dfnn3 | ⊢ ℕ = ∩ {𝑥 ∣ (𝑥 ⊆ ℝ ∧ 1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq2 2828 | . . . 4 ⊢ (𝑥 = 𝑧 → (1 ∈ 𝑥 ↔ 1 ∈ 𝑧)) | |
2 | eleq2 2828 | . . . . 5 ⊢ (𝑥 = 𝑧 → ((𝑦 + 1) ∈ 𝑥 ↔ (𝑦 + 1) ∈ 𝑧)) | |
3 | 2 | raleqbi1dv 3285 | . . . 4 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥 ↔ ∀𝑦 ∈ 𝑧 (𝑦 + 1) ∈ 𝑧)) |
4 | 1, 3 | anbi12d 749 | . . 3 ⊢ (𝑥 = 𝑧 → ((1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥) ↔ (1 ∈ 𝑧 ∧ ∀𝑦 ∈ 𝑧 (𝑦 + 1) ∈ 𝑧))) |
5 | dfnn2 11245 | . . . . 5 ⊢ ℕ = ∩ {𝑧 ∣ (1 ∈ 𝑧 ∧ ∀𝑦 ∈ 𝑧 (𝑦 + 1) ∈ 𝑧)} | |
6 | 5 | eqeq2i 2772 | . . . 4 ⊢ (𝑥 = ℕ ↔ 𝑥 = ∩ {𝑧 ∣ (1 ∈ 𝑧 ∧ ∀𝑦 ∈ 𝑧 (𝑦 + 1) ∈ 𝑧)}) |
7 | eleq2 2828 | . . . . 5 ⊢ (𝑥 = ℕ → (1 ∈ 𝑥 ↔ 1 ∈ ℕ)) | |
8 | eleq2 2828 | . . . . . 6 ⊢ (𝑥 = ℕ → ((𝑦 + 1) ∈ 𝑥 ↔ (𝑦 + 1) ∈ ℕ)) | |
9 | 8 | raleqbi1dv 3285 | . . . . 5 ⊢ (𝑥 = ℕ → (∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥 ↔ ∀𝑦 ∈ ℕ (𝑦 + 1) ∈ ℕ)) |
10 | 7, 9 | anbi12d 749 | . . . 4 ⊢ (𝑥 = ℕ → ((1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥) ↔ (1 ∈ ℕ ∧ ∀𝑦 ∈ ℕ (𝑦 + 1) ∈ ℕ))) |
11 | 6, 10 | sylbir 225 | . . 3 ⊢ (𝑥 = ∩ {𝑧 ∣ (1 ∈ 𝑧 ∧ ∀𝑦 ∈ 𝑧 (𝑦 + 1) ∈ 𝑧)} → ((1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥) ↔ (1 ∈ ℕ ∧ ∀𝑦 ∈ ℕ (𝑦 + 1) ∈ ℕ))) |
12 | nnssre 11236 | . . . . 5 ⊢ ℕ ⊆ ℝ | |
13 | 5, 12 | eqsstr3i 3777 | . . . 4 ⊢ ∩ {𝑧 ∣ (1 ∈ 𝑧 ∧ ∀𝑦 ∈ 𝑧 (𝑦 + 1) ∈ 𝑧)} ⊆ ℝ |
14 | 1nn 11243 | . . . . 5 ⊢ 1 ∈ ℕ | |
15 | peano2nn 11244 | . . . . . 6 ⊢ (𝑦 ∈ ℕ → (𝑦 + 1) ∈ ℕ) | |
16 | 15 | rgen 3060 | . . . . 5 ⊢ ∀𝑦 ∈ ℕ (𝑦 + 1) ∈ ℕ |
17 | 14, 16 | pm3.2i 470 | . . . 4 ⊢ (1 ∈ ℕ ∧ ∀𝑦 ∈ ℕ (𝑦 + 1) ∈ ℕ) |
18 | 13, 17 | pm3.2i 470 | . . 3 ⊢ (∩ {𝑧 ∣ (1 ∈ 𝑧 ∧ ∀𝑦 ∈ 𝑧 (𝑦 + 1) ∈ 𝑧)} ⊆ ℝ ∧ (1 ∈ ℕ ∧ ∀𝑦 ∈ ℕ (𝑦 + 1) ∈ ℕ)) |
19 | 4, 11, 18 | intabs 4974 | . 2 ⊢ ∩ {𝑥 ∣ (𝑥 ⊆ ℝ ∧ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥))} = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} |
20 | 3anass 1081 | . . . 4 ⊢ ((𝑥 ⊆ ℝ ∧ 1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥) ↔ (𝑥 ⊆ ℝ ∧ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥))) | |
21 | 20 | abbii 2877 | . . 3 ⊢ {𝑥 ∣ (𝑥 ⊆ ℝ ∧ 1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} = {𝑥 ∣ (𝑥 ⊆ ℝ ∧ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥))} |
22 | 21 | inteqi 4631 | . 2 ⊢ ∩ {𝑥 ∣ (𝑥 ⊆ ℝ ∧ 1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} = ∩ {𝑥 ∣ (𝑥 ⊆ ℝ ∧ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥))} |
23 | dfnn2 11245 | . 2 ⊢ ℕ = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} | |
24 | 19, 22, 23 | 3eqtr4ri 2793 | 1 ⊢ ℕ = ∩ {𝑥 ∣ (𝑥 ⊆ ℝ ∧ 1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 383 ∧ w3a 1072 = wceq 1632 ∈ wcel 2139 {cab 2746 ∀wral 3050 ⊆ wss 3715 ∩ cint 4627 (class class class)co 6814 ℝcr 10147 1c1 10149 + caddc 10151 ℕcn 11232 |
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-pow 4992 ax-pr 5055 ax-un 7115 ax-cnex 10204 ax-resscn 10205 ax-1cn 10206 ax-icn 10207 ax-addcl 10208 ax-addrcl 10209 ax-mulcl 10210 ax-mulrcl 10211 ax-i2m1 10216 ax-1ne0 10217 ax-rrecex 10220 ax-cnre 10221 |
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-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 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-int 4628 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-ov 6817 df-om 7232 df-wrecs 7577 df-recs 7638 df-rdg 7676 df-nn 11233 |
This theorem is referenced by: (None) |
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