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Mirrors > Home > ILE Home > Th. List > nnindnn | GIF version |
Description: Principle of Mathematical Induction (inference schema). This is a counterpart to nnind 8729 designed for real number axioms which involve natural numbers (notably, axcaucvg 7701). (Contributed by Jim Kingdon, 14-Jul-2021.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nntopi.n | ⊢ 𝑁 = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} |
nnindnn.1 | ⊢ (𝑧 = 1 → (𝜑 ↔ 𝜓)) |
nnindnn.y | ⊢ (𝑧 = 𝑘 → (𝜑 ↔ 𝜒)) |
nnindnn.y1 | ⊢ (𝑧 = (𝑘 + 1) → (𝜑 ↔ 𝜃)) |
nnindnn.a | ⊢ (𝑧 = 𝐴 → (𝜑 ↔ 𝜏)) |
nnindnn.basis | ⊢ 𝜓 |
nnindnn.step | ⊢ (𝑘 ∈ 𝑁 → (𝜒 → 𝜃)) |
Ref | Expression |
---|---|
nnindnn | ⊢ (𝐴 ∈ 𝑁 → 𝜏) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nntopi.n | . . . . . . 7 ⊢ 𝑁 = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} | |
2 | 1 | peano1nnnn 7653 | . . . . . 6 ⊢ 1 ∈ 𝑁 |
3 | nnindnn.basis | . . . . . 6 ⊢ 𝜓 | |
4 | nnindnn.1 | . . . . . . 7 ⊢ (𝑧 = 1 → (𝜑 ↔ 𝜓)) | |
5 | 4 | elrab 2835 | . . . . . 6 ⊢ (1 ∈ {𝑧 ∈ 𝑁 ∣ 𝜑} ↔ (1 ∈ 𝑁 ∧ 𝜓)) |
6 | 2, 3, 5 | mpbir2an 926 | . . . . 5 ⊢ 1 ∈ {𝑧 ∈ 𝑁 ∣ 𝜑} |
7 | elrabi 2832 | . . . . . . 7 ⊢ (𝑘 ∈ {𝑧 ∈ 𝑁 ∣ 𝜑} → 𝑘 ∈ 𝑁) | |
8 | 1 | peano2nnnn 7654 | . . . . . . . . . 10 ⊢ (𝑘 ∈ 𝑁 → (𝑘 + 1) ∈ 𝑁) |
9 | 8 | a1d 22 | . . . . . . . . 9 ⊢ (𝑘 ∈ 𝑁 → (𝑘 ∈ 𝑁 → (𝑘 + 1) ∈ 𝑁)) |
10 | nnindnn.step | . . . . . . . . 9 ⊢ (𝑘 ∈ 𝑁 → (𝜒 → 𝜃)) | |
11 | 9, 10 | anim12d 333 | . . . . . . . 8 ⊢ (𝑘 ∈ 𝑁 → ((𝑘 ∈ 𝑁 ∧ 𝜒) → ((𝑘 + 1) ∈ 𝑁 ∧ 𝜃))) |
12 | nnindnn.y | . . . . . . . . 9 ⊢ (𝑧 = 𝑘 → (𝜑 ↔ 𝜒)) | |
13 | 12 | elrab 2835 | . . . . . . . 8 ⊢ (𝑘 ∈ {𝑧 ∈ 𝑁 ∣ 𝜑} ↔ (𝑘 ∈ 𝑁 ∧ 𝜒)) |
14 | nnindnn.y1 | . . . . . . . . 9 ⊢ (𝑧 = (𝑘 + 1) → (𝜑 ↔ 𝜃)) | |
15 | 14 | elrab 2835 | . . . . . . . 8 ⊢ ((𝑘 + 1) ∈ {𝑧 ∈ 𝑁 ∣ 𝜑} ↔ ((𝑘 + 1) ∈ 𝑁 ∧ 𝜃)) |
16 | 11, 13, 15 | 3imtr4g 204 | . . . . . . 7 ⊢ (𝑘 ∈ 𝑁 → (𝑘 ∈ {𝑧 ∈ 𝑁 ∣ 𝜑} → (𝑘 + 1) ∈ {𝑧 ∈ 𝑁 ∣ 𝜑})) |
17 | 7, 16 | mpcom 36 | . . . . . 6 ⊢ (𝑘 ∈ {𝑧 ∈ 𝑁 ∣ 𝜑} → (𝑘 + 1) ∈ {𝑧 ∈ 𝑁 ∣ 𝜑}) |
18 | 17 | rgen 2483 | . . . . 5 ⊢ ∀𝑘 ∈ {𝑧 ∈ 𝑁 ∣ 𝜑} (𝑘 + 1) ∈ {𝑧 ∈ 𝑁 ∣ 𝜑} |
19 | 1 | peano5nnnn 7693 | . . . . 5 ⊢ ((1 ∈ {𝑧 ∈ 𝑁 ∣ 𝜑} ∧ ∀𝑘 ∈ {𝑧 ∈ 𝑁 ∣ 𝜑} (𝑘 + 1) ∈ {𝑧 ∈ 𝑁 ∣ 𝜑}) → 𝑁 ⊆ {𝑧 ∈ 𝑁 ∣ 𝜑}) |
20 | 6, 18, 19 | mp2an 422 | . . . 4 ⊢ 𝑁 ⊆ {𝑧 ∈ 𝑁 ∣ 𝜑} |
21 | 20 | sseli 3088 | . . 3 ⊢ (𝐴 ∈ 𝑁 → 𝐴 ∈ {𝑧 ∈ 𝑁 ∣ 𝜑}) |
22 | nnindnn.a | . . . 4 ⊢ (𝑧 = 𝐴 → (𝜑 ↔ 𝜏)) | |
23 | 22 | elrab 2835 | . . 3 ⊢ (𝐴 ∈ {𝑧 ∈ 𝑁 ∣ 𝜑} ↔ (𝐴 ∈ 𝑁 ∧ 𝜏)) |
24 | 21, 23 | sylib 121 | . 2 ⊢ (𝐴 ∈ 𝑁 → (𝐴 ∈ 𝑁 ∧ 𝜏)) |
25 | 24 | simprd 113 | 1 ⊢ (𝐴 ∈ 𝑁 → 𝜏) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1331 ∈ wcel 1480 {cab 2123 ∀wral 2414 {crab 2418 ⊆ wss 3066 ∩ cint 3766 (class class class)co 5767 1c1 7614 + caddc 7616 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-eprel 4206 df-id 4210 df-po 4213 df-iso 4214 df-iord 4283 df-on 4285 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-irdg 6260 df-1o 6306 df-2o 6307 df-oadd 6310 df-omul 6311 df-er 6422 df-ec 6424 df-qs 6428 df-ni 7105 df-pli 7106 df-mi 7107 df-lti 7108 df-plpq 7145 df-mpq 7146 df-enq 7148 df-nqqs 7149 df-plqqs 7150 df-mqqs 7151 df-1nqqs 7152 df-rq 7153 df-ltnqqs 7154 df-enq0 7225 df-nq0 7226 df-0nq0 7227 df-plq0 7228 df-mq0 7229 df-inp 7267 df-i1p 7268 df-iplp 7269 df-enr 7527 df-nr 7528 df-plr 7529 df-0r 7532 df-1r 7533 df-c 7619 df-1 7621 df-r 7623 df-add 7624 |
This theorem is referenced by: nntopi 7695 |
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