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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 8864 designed for real number axioms which involve natural numbers (notably, axcaucvg 7832). (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 7784 | . . . . . 6 ⊢ 1 ∈ 𝑁 |
3 | nnindnn.basis | . . . . . 6 ⊢ 𝜓 | |
4 | nnindnn.1 | . . . . . . 7 ⊢ (𝑧 = 1 → (𝜑 ↔ 𝜓)) | |
5 | 4 | elrab 2877 | . . . . . 6 ⊢ (1 ∈ {𝑧 ∈ 𝑁 ∣ 𝜑} ↔ (1 ∈ 𝑁 ∧ 𝜓)) |
6 | 2, 3, 5 | mpbir2an 931 | . . . . 5 ⊢ 1 ∈ {𝑧 ∈ 𝑁 ∣ 𝜑} |
7 | elrabi 2874 | . . . . . . 7 ⊢ (𝑘 ∈ {𝑧 ∈ 𝑁 ∣ 𝜑} → 𝑘 ∈ 𝑁) | |
8 | 1 | peano2nnnn 7785 | . . . . . . . . . 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 2877 | . . . . . . . 8 ⊢ (𝑘 ∈ {𝑧 ∈ 𝑁 ∣ 𝜑} ↔ (𝑘 ∈ 𝑁 ∧ 𝜒)) |
14 | nnindnn.y1 | . . . . . . . . 9 ⊢ (𝑧 = (𝑘 + 1) → (𝜑 ↔ 𝜃)) | |
15 | 14 | elrab 2877 | . . . . . . . 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 2517 | . . . . 5 ⊢ ∀𝑘 ∈ {𝑧 ∈ 𝑁 ∣ 𝜑} (𝑘 + 1) ∈ {𝑧 ∈ 𝑁 ∣ 𝜑} |
19 | 1 | peano5nnnn 7824 | . . . . 5 ⊢ ((1 ∈ {𝑧 ∈ 𝑁 ∣ 𝜑} ∧ ∀𝑘 ∈ {𝑧 ∈ 𝑁 ∣ 𝜑} (𝑘 + 1) ∈ {𝑧 ∈ 𝑁 ∣ 𝜑}) → 𝑁 ⊆ {𝑧 ∈ 𝑁 ∣ 𝜑}) |
20 | 6, 18, 19 | mp2an 423 | . . . 4 ⊢ 𝑁 ⊆ {𝑧 ∈ 𝑁 ∣ 𝜑} |
21 | 20 | sseli 3133 | . . 3 ⊢ (𝐴 ∈ 𝑁 → 𝐴 ∈ {𝑧 ∈ 𝑁 ∣ 𝜑}) |
22 | nnindnn.a | . . . 4 ⊢ (𝑧 = 𝐴 → (𝜑 ↔ 𝜏)) | |
23 | 22 | elrab 2877 | . . 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 1342 ∈ wcel 2135 {cab 2150 ∀wral 2442 {crab 2446 ⊆ wss 3111 ∩ cint 3818 (class class class)co 5836 1c1 7745 + caddc 7747 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-eprel 4261 df-id 4265 df-po 4268 df-iso 4269 df-iord 4338 df-on 4340 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-irdg 6329 df-1o 6375 df-2o 6376 df-oadd 6379 df-omul 6380 df-er 6492 df-ec 6494 df-qs 6498 df-ni 7236 df-pli 7237 df-mi 7238 df-lti 7239 df-plpq 7276 df-mpq 7277 df-enq 7279 df-nqqs 7280 df-plqqs 7281 df-mqqs 7282 df-1nqqs 7283 df-rq 7284 df-ltnqqs 7285 df-enq0 7356 df-nq0 7357 df-0nq0 7358 df-plq0 7359 df-mq0 7360 df-inp 7398 df-i1p 7399 df-iplp 7400 df-enr 7658 df-nr 7659 df-plr 7660 df-0r 7663 df-1r 7664 df-c 7750 df-1 7752 df-r 7754 df-add 7755 |
This theorem is referenced by: nntopi 7826 |
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