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| Mirrors > Home > MPE Home > Th. List > Mathboxes > nnindf | Structured version Visualization version GIF version | ||
| Description: Principle of Mathematical Induction, using a bound-variable hypothesis instead of distinct variables. (Contributed by Thierry Arnoux, 6-May-2018.) |
| Ref | Expression |
|---|---|
| nnindf.x | ⊢ Ⅎ𝑦𝜑 |
| nnindf.1 | ⊢ (𝑥 = 1 → (𝜑 ↔ 𝜓)) |
| nnindf.2 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) |
| nnindf.3 | ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜃)) |
| nnindf.4 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏)) |
| nnindf.5 | ⊢ 𝜓 |
| nnindf.6 | ⊢ (𝑦 ∈ ℕ → (𝜒 → 𝜃)) |
| Ref | Expression |
|---|---|
| nnindf | ⊢ (𝐴 ∈ ℕ → 𝜏) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 1nn 12277 | . . . . . 6 ⊢ 1 ∈ ℕ | |
| 2 | nnindf.5 | . . . . . 6 ⊢ 𝜓 | |
| 3 | nnindf.1 | . . . . . . 7 ⊢ (𝑥 = 1 → (𝜑 ↔ 𝜓)) | |
| 4 | 3 | elrab 3692 | . . . . . 6 ⊢ (1 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ (1 ∈ ℕ ∧ 𝜓)) |
| 5 | 1, 2, 4 | mpbir2an 711 | . . . . 5 ⊢ 1 ∈ {𝑥 ∈ ℕ ∣ 𝜑} |
| 6 | elrabi 3687 | . . . . . . . 8 ⊢ (𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} → 𝑦 ∈ ℕ) | |
| 7 | peano2nn 12278 | . . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ → (𝑦 + 1) ∈ ℕ) | |
| 8 | 7 | a1d 25 | . . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → (𝑦 ∈ ℕ → (𝑦 + 1) ∈ ℕ)) |
| 9 | nnindf.6 | . . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → (𝜒 → 𝜃)) | |
| 10 | 8, 9 | anim12d 609 | . . . . . . . . 9 ⊢ (𝑦 ∈ ℕ → ((𝑦 ∈ ℕ ∧ 𝜒) → ((𝑦 + 1) ∈ ℕ ∧ 𝜃))) |
| 11 | nnindf.2 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) | |
| 12 | 11 | elrab 3692 | . . . . . . . . 9 ⊢ (𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ (𝑦 ∈ ℕ ∧ 𝜒)) |
| 13 | nnindf.3 | . . . . . . . . . 10 ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜃)) | |
| 14 | 13 | elrab 3692 | . . . . . . . . 9 ⊢ ((𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ ((𝑦 + 1) ∈ ℕ ∧ 𝜃)) |
| 15 | 10, 12, 14 | 3imtr4g 296 | . . . . . . . 8 ⊢ (𝑦 ∈ ℕ → (𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} → (𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑})) |
| 16 | 6, 15 | mpcom 38 | . . . . . . 7 ⊢ (𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} → (𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑}) |
| 17 | 16 | rgen 3063 | . . . . . 6 ⊢ ∀𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} (𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑} |
| 18 | nnindf.x | . . . . . . . 8 ⊢ Ⅎ𝑦𝜑 | |
| 19 | nfcv 2905 | . . . . . . . 8 ⊢ Ⅎ𝑦ℕ | |
| 20 | 18, 19 | nfrabw 3475 | . . . . . . 7 ⊢ Ⅎ𝑦{𝑥 ∈ ℕ ∣ 𝜑} |
| 21 | nfcv 2905 | . . . . . . 7 ⊢ Ⅎ𝑤{𝑥 ∈ ℕ ∣ 𝜑} | |
| 22 | nfv 1914 | . . . . . . 7 ⊢ Ⅎ𝑤(𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑} | |
| 23 | 20 | nfel2 2924 | . . . . . . 7 ⊢ Ⅎ𝑦(𝑤 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑} |
| 24 | oveq1 7438 | . . . . . . . 8 ⊢ (𝑦 = 𝑤 → (𝑦 + 1) = (𝑤 + 1)) | |
| 25 | 24 | eleq1d 2826 | . . . . . . 7 ⊢ (𝑦 = 𝑤 → ((𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ (𝑤 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑})) |
| 26 | 20, 21, 22, 23, 25 | cbvralfw 3304 | . . . . . 6 ⊢ (∀𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} (𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ ∀𝑤 ∈ {𝑥 ∈ ℕ ∣ 𝜑} (𝑤 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑}) |
| 27 | 17, 26 | mpbi 230 | . . . . 5 ⊢ ∀𝑤 ∈ {𝑥 ∈ ℕ ∣ 𝜑} (𝑤 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑} |
| 28 | peano5nni 12269 | . . . . 5 ⊢ ((1 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ∧ ∀𝑤 ∈ {𝑥 ∈ ℕ ∣ 𝜑} (𝑤 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑}) → ℕ ⊆ {𝑥 ∈ ℕ ∣ 𝜑}) | |
| 29 | 5, 27, 28 | mp2an 692 | . . . 4 ⊢ ℕ ⊆ {𝑥 ∈ ℕ ∣ 𝜑} |
| 30 | 29 | sseli 3979 | . . 3 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ {𝑥 ∈ ℕ ∣ 𝜑}) |
| 31 | nnindf.4 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏)) | |
| 32 | 31 | elrab 3692 | . . 3 ⊢ (𝐴 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ (𝐴 ∈ ℕ ∧ 𝜏)) |
| 33 | 30, 32 | sylib 218 | . 2 ⊢ (𝐴 ∈ ℕ → (𝐴 ∈ ℕ ∧ 𝜏)) |
| 34 | 33 | simprd 495 | 1 ⊢ (𝐴 ∈ ℕ → 𝜏) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2108 ∀wral 3061 {crab 3436 ⊆ wss 3951 (class class class)co 7431 1c1 11156 + caddc 11158 ℕcn 12266 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 ax-1cn 11213 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-nn 12267 |
| This theorem is referenced by: nn0min 32822 |
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