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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 3336 | . . . 4 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥 ↔ ∀𝑦 ∈ 𝑧 (𝑦 + 1) ∈ 𝑧)) |
4 | 1, 3 | anbi12d 632 | . . 3 ⊢ (𝑥 = 𝑧 → ((1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥) ↔ (1 ∈ 𝑧 ∧ ∀𝑦 ∈ 𝑧 (𝑦 + 1) ∈ 𝑧))) |
5 | dfnn2 12277 | . . . . 5 ⊢ ℕ = ∩ {𝑧 ∣ (1 ∈ 𝑧 ∧ ∀𝑦 ∈ 𝑧 (𝑦 + 1) ∈ 𝑧)} | |
6 | 5 | eqeq2i 2748 | . . . 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 3336 | . . . . 5 ⊢ (𝑥 = ℕ → (∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥 ↔ ∀𝑦 ∈ ℕ (𝑦 + 1) ∈ ℕ)) |
10 | 7, 9 | anbi12d 632 | . . . 4 ⊢ (𝑥 = ℕ → ((1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥) ↔ (1 ∈ ℕ ∧ ∀𝑦 ∈ ℕ (𝑦 + 1) ∈ ℕ))) |
11 | 6, 10 | sylbir 235 | . . 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 12268 | . . . . 5 ⊢ ℕ ⊆ ℝ | |
13 | 5, 12 | eqsstrri 4031 | . . . 4 ⊢ ∩ {𝑧 ∣ (1 ∈ 𝑧 ∧ ∀𝑦 ∈ 𝑧 (𝑦 + 1) ∈ 𝑧)} ⊆ ℝ |
14 | 1nn 12275 | . . . . 5 ⊢ 1 ∈ ℕ | |
15 | peano2nn 12276 | . . . . . 6 ⊢ (𝑦 ∈ ℕ → (𝑦 + 1) ∈ ℕ) | |
16 | 15 | rgen 3061 | . . . . 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 5355 | . 2 ⊢ ∩ {𝑥 ∣ (𝑥 ⊆ ℝ ∧ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥))} = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} |
20 | 3anass 1094 | . . . 4 ⊢ ((𝑥 ⊆ ℝ ∧ 1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥) ↔ (𝑥 ⊆ ℝ ∧ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥))) | |
21 | 20 | abbii 2807 | . . 3 ⊢ {𝑥 ∣ (𝑥 ⊆ ℝ ∧ 1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} = {𝑥 ∣ (𝑥 ⊆ ℝ ∧ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥))} |
22 | 21 | inteqi 4955 | . 2 ⊢ ∩ {𝑥 ∣ (𝑥 ⊆ ℝ ∧ 1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} = ∩ {𝑥 ∣ (𝑥 ⊆ ℝ ∧ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥))} |
23 | dfnn2 12277 | . 2 ⊢ ℕ = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} | |
24 | 19, 22, 23 | 3eqtr4ri 2774 | 1 ⊢ ℕ = ∩ {𝑥 ∣ (𝑥 ⊆ ℝ ∧ 1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 {cab 2712 ∀wral 3059 ⊆ wss 3963 ∩ cint 4951 (class class class)co 7431 ℝcr 11152 1c1 11154 + caddc 11156 ℕcn 12264 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-i2m1 11221 ax-1ne0 11222 ax-rrecex 11225 ax-cnre 11226 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-nn 12265 |
This theorem is referenced by: (None) |
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