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Mirrors > Home > ILE Home > Th. List > 1nn | GIF version |
Description: Peano postulate: 1 is a positive integer. (Contributed by NM, 11-Jan-1997.) |
Ref | Expression |
---|---|
1nn | ⊢ 1 ∈ ℕ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfnn2 7916 | . . . 4 ⊢ ℕ = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} | |
2 | 1 | eleq2i 2104 | . . 3 ⊢ (1 ∈ ℕ ↔ 1 ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}) |
3 | 1re 7026 | . . . 4 ⊢ 1 ∈ ℝ | |
4 | elintg 3623 | . . . 4 ⊢ (1 ∈ ℝ → (1 ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} ↔ ∀𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}1 ∈ 𝑧)) | |
5 | 3, 4 | ax-mp 7 | . . 3 ⊢ (1 ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} ↔ ∀𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}1 ∈ 𝑧) |
6 | 2, 5 | bitri 173 | . 2 ⊢ (1 ∈ ℕ ↔ ∀𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}1 ∈ 𝑧) |
7 | vex 2560 | . . . 4 ⊢ 𝑧 ∈ V | |
8 | eleq2 2101 | . . . . 5 ⊢ (𝑥 = 𝑧 → (1 ∈ 𝑥 ↔ 1 ∈ 𝑧)) | |
9 | eleq2 2101 | . . . . . 6 ⊢ (𝑥 = 𝑧 → ((𝑦 + 1) ∈ 𝑥 ↔ (𝑦 + 1) ∈ 𝑧)) | |
10 | 9 | raleqbi1dv 2513 | . . . . 5 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥 ↔ ∀𝑦 ∈ 𝑧 (𝑦 + 1) ∈ 𝑧)) |
11 | 8, 10 | anbi12d 442 | . . . 4 ⊢ (𝑥 = 𝑧 → ((1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥) ↔ (1 ∈ 𝑧 ∧ ∀𝑦 ∈ 𝑧 (𝑦 + 1) ∈ 𝑧))) |
12 | 7, 11 | elab 2687 | . . 3 ⊢ (𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} ↔ (1 ∈ 𝑧 ∧ ∀𝑦 ∈ 𝑧 (𝑦 + 1) ∈ 𝑧)) |
13 | 12 | simplbi 259 | . 2 ⊢ (𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} → 1 ∈ 𝑧) |
14 | 6, 13 | mprgbir 2379 | 1 ⊢ 1 ∈ ℕ |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ↔ wb 98 ∈ wcel 1393 {cab 2026 ∀wral 2306 ∩ cint 3615 (class class class)co 5512 ℝcr 6888 1c1 6890 + caddc 6892 ℕcn 7914 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-1re 6978 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-v 2559 df-int 3616 df-inn 7915 |
This theorem is referenced by: nnind 7930 nn1suc 7933 2nn 8077 1nn0 8197 nn0p1nn 8221 1z 8271 neg1z 8277 elz2 8312 nneoor 8340 indstr 8536 elnn1uz2 8544 zq 8561 qreccl 8576 expivallem 9256 exp1 9261 nnexpcl 9268 expnbnd 9372 resqrexlemf1 9606 resqrexlemcalc3 9614 resqrexlemnmsq 9615 resqrexlemnm 9616 resqrexlemcvg 9617 resqrexlemglsq 9620 resqrexlemga 9621 |
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