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Mirrors > Home > ILE Home > Th. List > peano1nnnn | GIF version |
Description: One is an element of ℕ. This is a counterpart to 1nn 8755 designed for real number axioms which involve natural numbers (notably, axcaucvg 7732). (Contributed by Jim Kingdon, 14-Jul-2021.) (New usage is discouraged.) |
Ref | Expression |
---|---|
peano1nnnn.n | ⊢ 𝑁 = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} |
Ref | Expression |
---|---|
peano1nnnn | ⊢ 1 ∈ 𝑁 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | peano1nnnn.n | . . . 4 ⊢ 𝑁 = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} | |
2 | 1 | eleq2i 2207 | . . 3 ⊢ (1 ∈ 𝑁 ↔ 1 ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}) |
3 | df-1 7652 | . . . . 5 ⊢ 1 = 〈1R, 0R〉 | |
4 | 1sr 7583 | . . . . . 6 ⊢ 1R ∈ R | |
5 | opelreal 7659 | . . . . . 6 ⊢ (〈1R, 0R〉 ∈ ℝ ↔ 1R ∈ R) | |
6 | 4, 5 | mpbir 145 | . . . . 5 ⊢ 〈1R, 0R〉 ∈ ℝ |
7 | 3, 6 | eqeltri 2213 | . . . 4 ⊢ 1 ∈ ℝ |
8 | elintg 3787 | . . . 4 ⊢ (1 ∈ ℝ → (1 ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} ↔ ∀𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}1 ∈ 𝑧)) | |
9 | 7, 8 | ax-mp 5 | . . 3 ⊢ (1 ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} ↔ ∀𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}1 ∈ 𝑧) |
10 | 2, 9 | bitri 183 | . 2 ⊢ (1 ∈ 𝑁 ↔ ∀𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}1 ∈ 𝑧) |
11 | vex 2692 | . . . 4 ⊢ 𝑧 ∈ V | |
12 | eleq2 2204 | . . . . 5 ⊢ (𝑥 = 𝑧 → (1 ∈ 𝑥 ↔ 1 ∈ 𝑧)) | |
13 | eleq2 2204 | . . . . . 6 ⊢ (𝑥 = 𝑧 → ((𝑦 + 1) ∈ 𝑥 ↔ (𝑦 + 1) ∈ 𝑧)) | |
14 | 13 | raleqbi1dv 2637 | . . . . 5 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥 ↔ ∀𝑦 ∈ 𝑧 (𝑦 + 1) ∈ 𝑧)) |
15 | 12, 14 | anbi12d 465 | . . . 4 ⊢ (𝑥 = 𝑧 → ((1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥) ↔ (1 ∈ 𝑧 ∧ ∀𝑦 ∈ 𝑧 (𝑦 + 1) ∈ 𝑧))) |
16 | 11, 15 | elab 2832 | . . 3 ⊢ (𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} ↔ (1 ∈ 𝑧 ∧ ∀𝑦 ∈ 𝑧 (𝑦 + 1) ∈ 𝑧)) |
17 | 16 | simplbi 272 | . 2 ⊢ (𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} → 1 ∈ 𝑧) |
18 | 10, 17 | mprgbir 2493 | 1 ⊢ 1 ∈ 𝑁 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 = wceq 1332 ∈ wcel 1481 {cab 2126 ∀wral 2417 〈cop 3535 ∩ cint 3779 (class class class)co 5782 Rcnr 7129 0Rc0r 7130 1Rc1r 7131 ℝcr 7643 1c1 7645 + caddc 7647 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-eprel 4219 df-id 4223 df-po 4226 df-iso 4227 df-iord 4296 df-on 4298 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-irdg 6275 df-1o 6321 df-2o 6322 df-oadd 6325 df-omul 6326 df-er 6437 df-ec 6439 df-qs 6443 df-ni 7136 df-pli 7137 df-mi 7138 df-lti 7139 df-plpq 7176 df-mpq 7177 df-enq 7179 df-nqqs 7180 df-plqqs 7181 df-mqqs 7182 df-1nqqs 7183 df-rq 7184 df-ltnqqs 7185 df-enq0 7256 df-nq0 7257 df-0nq0 7258 df-plq0 7259 df-mq0 7260 df-inp 7298 df-i1p 7299 df-iplp 7300 df-enr 7558 df-nr 7559 df-0r 7563 df-1r 7564 df-1 7652 df-r 7654 |
This theorem is referenced by: nnindnn 7725 |
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