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Mirrors > Home > ILE Home > Th. List > peano1 | GIF version |
Description: Zero is a natural number. One of Peano's five postulates for arithmetic. Proposition 7.30(1) of [TakeutiZaring] p. 42. (Contributed by NM, 15-May-1994.) |
Ref | Expression |
---|---|
peano1 | ⊢ ∅ ∈ ω |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 3905 | . . . 4 ⊢ ∅ ∈ V | |
2 | 1 | elint 3642 | . . 3 ⊢ (∅ ∈ ∩ {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥 ∈ 𝑦 suc 𝑥 ∈ 𝑦)} ↔ ∀𝑧(𝑧 ∈ {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥 ∈ 𝑦 suc 𝑥 ∈ 𝑦)} → ∅ ∈ 𝑧)) |
3 | df-clab 2068 | . . . 4 ⊢ (𝑧 ∈ {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥 ∈ 𝑦 suc 𝑥 ∈ 𝑦)} ↔ [𝑧 / 𝑦](∅ ∈ 𝑦 ∧ ∀𝑥 ∈ 𝑦 suc 𝑥 ∈ 𝑦)) | |
4 | simpl 107 | . . . . . 6 ⊢ ((∅ ∈ 𝑦 ∧ ∀𝑥 ∈ 𝑦 suc 𝑥 ∈ 𝑦) → ∅ ∈ 𝑦) | |
5 | 4 | sbimi 1687 | . . . . 5 ⊢ ([𝑧 / 𝑦](∅ ∈ 𝑦 ∧ ∀𝑥 ∈ 𝑦 suc 𝑥 ∈ 𝑦) → [𝑧 / 𝑦]∅ ∈ 𝑦) |
6 | clelsb4 2184 | . . . . 5 ⊢ ([𝑧 / 𝑦]∅ ∈ 𝑦 ↔ ∅ ∈ 𝑧) | |
7 | 5, 6 | sylib 120 | . . . 4 ⊢ ([𝑧 / 𝑦](∅ ∈ 𝑦 ∧ ∀𝑥 ∈ 𝑦 suc 𝑥 ∈ 𝑦) → ∅ ∈ 𝑧) |
8 | 3, 7 | sylbi 119 | . . 3 ⊢ (𝑧 ∈ {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥 ∈ 𝑦 suc 𝑥 ∈ 𝑦)} → ∅ ∈ 𝑧) |
9 | 2, 8 | mpgbir 1382 | . 2 ⊢ ∅ ∈ ∩ {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥 ∈ 𝑦 suc 𝑥 ∈ 𝑦)} |
10 | dfom3 4333 | . 2 ⊢ ω = ∩ {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥 ∈ 𝑦 suc 𝑥 ∈ 𝑦)} | |
11 | 9, 10 | eleqtrri 2154 | 1 ⊢ ∅ ∈ ω |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ∈ wcel 1433 [wsb 1685 {cab 2067 ∀wral 2348 ∅c0 3251 ∩ cint 3636 suc csuc 4120 ωcom 4331 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-nul 3904 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-dif 2975 df-nul 3252 df-int 3637 df-iom 4332 |
This theorem is referenced by: peano5 4339 limom 4354 nnregexmid 4360 frec0g 6006 frecrdg 6015 oa1suc 6070 nna0r 6080 nnm0r 6081 nnmcl 6083 nnmsucr 6090 1onn 6116 nnm1 6120 nnaordex 6123 nnawordex 6124 php5 6344 php5dom 6349 0fin 6368 findcard2 6373 findcard2s 6374 1lt2pi 6530 nq0m0r 6646 nq0a0 6647 prarloclem5 6690 frec2uzrand 9407 frecuzrdg0 9416 frecfzennn 9419 bj-nn0suc 10759 bj-nn0sucALT 10773 |
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