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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 8911 designed for real number axioms which involve natural numbers (notably, axcaucvg 7877). (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 7829 | . . . . . 6 ⊢ 1 ∈ 𝑁 |
3 | nnindnn.basis | . . . . . 6 ⊢ 𝜓 | |
4 | nnindnn.1 | . . . . . . 7 ⊢ (𝑧 = 1 → (𝜑 ↔ 𝜓)) | |
5 | 4 | elrab 2893 | . . . . . 6 ⊢ (1 ∈ {𝑧 ∈ 𝑁 ∣ 𝜑} ↔ (1 ∈ 𝑁 ∧ 𝜓)) |
6 | 2, 3, 5 | mpbir2an 942 | . . . . 5 ⊢ 1 ∈ {𝑧 ∈ 𝑁 ∣ 𝜑} |
7 | elrabi 2890 | . . . . . . 7 ⊢ (𝑘 ∈ {𝑧 ∈ 𝑁 ∣ 𝜑} → 𝑘 ∈ 𝑁) | |
8 | 1 | peano2nnnn 7830 | . . . . . . . . . 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 335 | . . . . . . . 8 ⊢ (𝑘 ∈ 𝑁 → ((𝑘 ∈ 𝑁 ∧ 𝜒) → ((𝑘 + 1) ∈ 𝑁 ∧ 𝜃))) |
12 | nnindnn.y | . . . . . . . . 9 ⊢ (𝑧 = 𝑘 → (𝜑 ↔ 𝜒)) | |
13 | 12 | elrab 2893 | . . . . . . . 8 ⊢ (𝑘 ∈ {𝑧 ∈ 𝑁 ∣ 𝜑} ↔ (𝑘 ∈ 𝑁 ∧ 𝜒)) |
14 | nnindnn.y1 | . . . . . . . . 9 ⊢ (𝑧 = (𝑘 + 1) → (𝜑 ↔ 𝜃)) | |
15 | 14 | elrab 2893 | . . . . . . . 8 ⊢ ((𝑘 + 1) ∈ {𝑧 ∈ 𝑁 ∣ 𝜑} ↔ ((𝑘 + 1) ∈ 𝑁 ∧ 𝜃)) |
16 | 11, 13, 15 | 3imtr4g 205 | . . . . . . 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 2530 | . . . . 5 ⊢ ∀𝑘 ∈ {𝑧 ∈ 𝑁 ∣ 𝜑} (𝑘 + 1) ∈ {𝑧 ∈ 𝑁 ∣ 𝜑} |
19 | 1 | peano5nnnn 7869 | . . . . 5 ⊢ ((1 ∈ {𝑧 ∈ 𝑁 ∣ 𝜑} ∧ ∀𝑘 ∈ {𝑧 ∈ 𝑁 ∣ 𝜑} (𝑘 + 1) ∈ {𝑧 ∈ 𝑁 ∣ 𝜑}) → 𝑁 ⊆ {𝑧 ∈ 𝑁 ∣ 𝜑}) |
20 | 6, 18, 19 | mp2an 426 | . . . 4 ⊢ 𝑁 ⊆ {𝑧 ∈ 𝑁 ∣ 𝜑} |
21 | 20 | sseli 3151 | . . 3 ⊢ (𝐴 ∈ 𝑁 → 𝐴 ∈ {𝑧 ∈ 𝑁 ∣ 𝜑}) |
22 | nnindnn.a | . . . 4 ⊢ (𝑧 = 𝐴 → (𝜑 ↔ 𝜏)) | |
23 | 22 | elrab 2893 | . . 3 ⊢ (𝐴 ∈ {𝑧 ∈ 𝑁 ∣ 𝜑} ↔ (𝐴 ∈ 𝑁 ∧ 𝜏)) |
24 | 21, 23 | sylib 122 | . 2 ⊢ (𝐴 ∈ 𝑁 → (𝐴 ∈ 𝑁 ∧ 𝜏)) |
25 | 24 | simprd 114 | 1 ⊢ (𝐴 ∈ 𝑁 → 𝜏) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1353 ∈ wcel 2148 {cab 2163 ∀wral 2455 {crab 2459 ⊆ wss 3129 ∩ cint 3842 (class class class)co 5868 1c1 7790 + caddc 7792 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 ax-iinf 4583 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-eprel 4285 df-id 4289 df-po 4292 df-iso 4293 df-iord 4362 df-on 4364 df-suc 4367 df-iom 4586 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-ima 4635 df-iota 5173 df-fun 5213 df-fn 5214 df-f 5215 df-f1 5216 df-fo 5217 df-f1o 5218 df-fv 5219 df-ov 5871 df-oprab 5872 df-mpo 5873 df-1st 6134 df-2nd 6135 df-recs 6299 df-irdg 6364 df-1o 6410 df-2o 6411 df-oadd 6414 df-omul 6415 df-er 6528 df-ec 6530 df-qs 6534 df-ni 7281 df-pli 7282 df-mi 7283 df-lti 7284 df-plpq 7321 df-mpq 7322 df-enq 7324 df-nqqs 7325 df-plqqs 7326 df-mqqs 7327 df-1nqqs 7328 df-rq 7329 df-ltnqqs 7330 df-enq0 7401 df-nq0 7402 df-0nq0 7403 df-plq0 7404 df-mq0 7405 df-inp 7443 df-i1p 7444 df-iplp 7445 df-enr 7703 df-nr 7704 df-plr 7705 df-0r 7708 df-1r 7709 df-c 7795 df-1 7797 df-r 7799 df-add 7800 |
This theorem is referenced by: nntopi 7871 |
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