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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 11636 | . . . . . 6 ⊢ 1 ∈ ℕ | |
2 | nnindf.5 | . . . . . 6 ⊢ 𝜓 | |
3 | nnindf.1 | . . . . . . 7 ⊢ (𝑥 = 1 → (𝜑 ↔ 𝜓)) | |
4 | 3 | elrab 3628 | . . . . . 6 ⊢ (1 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ (1 ∈ ℕ ∧ 𝜓)) |
5 | 1, 2, 4 | mpbir2an 710 | . . . . 5 ⊢ 1 ∈ {𝑥 ∈ ℕ ∣ 𝜑} |
6 | elrabi 3623 | . . . . . . . 8 ⊢ (𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} → 𝑦 ∈ ℕ) | |
7 | peano2nn 11637 | . . . . . . . . . . 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 611 | . . . . . . . . 9 ⊢ (𝑦 ∈ ℕ → ((𝑦 ∈ ℕ ∧ 𝜒) → ((𝑦 + 1) ∈ ℕ ∧ 𝜃))) |
11 | nnindf.2 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) | |
12 | 11 | elrab 3628 | . . . . . . . . 9 ⊢ (𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ (𝑦 ∈ ℕ ∧ 𝜒)) |
13 | nnindf.3 | . . . . . . . . . 10 ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜃)) | |
14 | 13 | elrab 3628 | . . . . . . . . 9 ⊢ ((𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ ((𝑦 + 1) ∈ ℕ ∧ 𝜃)) |
15 | 10, 12, 14 | 3imtr4g 299 | . . . . . . . 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 3116 | . . . . . 6 ⊢ ∀𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} (𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑} |
18 | nnindf.x | . . . . . . . 8 ⊢ Ⅎ𝑦𝜑 | |
19 | nfcv 2955 | . . . . . . . 8 ⊢ Ⅎ𝑦ℕ | |
20 | 18, 19 | nfrabw 3338 | . . . . . . 7 ⊢ Ⅎ𝑦{𝑥 ∈ ℕ ∣ 𝜑} |
21 | nfcv 2955 | . . . . . . 7 ⊢ Ⅎ𝑤{𝑥 ∈ ℕ ∣ 𝜑} | |
22 | nfv 1915 | . . . . . . 7 ⊢ Ⅎ𝑤(𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑} | |
23 | 20 | nfel2 2973 | . . . . . . 7 ⊢ Ⅎ𝑦(𝑤 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑} |
24 | oveq1 7142 | . . . . . . . 8 ⊢ (𝑦 = 𝑤 → (𝑦 + 1) = (𝑤 + 1)) | |
25 | 24 | eleq1d 2874 | . . . . . . 7 ⊢ (𝑦 = 𝑤 → ((𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ (𝑤 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑})) |
26 | 20, 21, 22, 23, 25 | cbvralfw 3382 | . . . . . 6 ⊢ (∀𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} (𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ ∀𝑤 ∈ {𝑥 ∈ ℕ ∣ 𝜑} (𝑤 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑}) |
27 | 17, 26 | mpbi 233 | . . . . 5 ⊢ ∀𝑤 ∈ {𝑥 ∈ ℕ ∣ 𝜑} (𝑤 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑} |
28 | peano5nni 11628 | . . . . 5 ⊢ ((1 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ∧ ∀𝑤 ∈ {𝑥 ∈ ℕ ∣ 𝜑} (𝑤 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑}) → ℕ ⊆ {𝑥 ∈ ℕ ∣ 𝜑}) | |
29 | 5, 27, 28 | mp2an 691 | . . . 4 ⊢ ℕ ⊆ {𝑥 ∈ ℕ ∣ 𝜑} |
30 | 29 | sseli 3911 | . . 3 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ {𝑥 ∈ ℕ ∣ 𝜑}) |
31 | nnindf.4 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏)) | |
32 | 31 | elrab 3628 | . . 3 ⊢ (𝐴 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ (𝐴 ∈ ℕ ∧ 𝜏)) |
33 | 30, 32 | sylib 221 | . 2 ⊢ (𝐴 ∈ ℕ → (𝐴 ∈ ℕ ∧ 𝜏)) |
34 | 33 | simprd 499 | 1 ⊢ (𝐴 ∈ ℕ → 𝜏) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 Ⅎwnf 1785 ∈ wcel 2111 ∀wral 3106 {crab 3110 ⊆ wss 3881 (class class class)co 7135 1c1 10527 + caddc 10529 ℕcn 11625 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-1cn 10584 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-nn 11626 |
This theorem is referenced by: nn0min 30562 |
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